Rangers continued their incredible run in the Europa League – but they were made to work for it by a spirited Braga side.
A brilliant brace from irrepressible captain James Tavernier saw them overturn a one-goal deficit from the first leg in Portugal.
They looked like continuing a march that is putting the continent on notice but defender David Carmo had other ideas as 10-man Braga stirred into life late on and forced extra time from nowhere.
But Kemar Roofe stepped up as the game clock ticked into triple figures to send Gers through.
And the prize for another thunderous Thursday night performance in Govan is a battle against RB Leipzig in the final four after the Germans saw off Atalanta hours earlier.
Giovanni van Bronckhorst sprung few surprises in his team tasked with turning the tie around, with a packed midfield including Aaron Ramsey's experience, John Lundstram's relentlessness and Ryan Jack's composure.
Braga braced themselves, knowing their opposition had the nervous energy of 50,000 souls to spur them on and in the knowledge they had a 1-0 lead to hang on to.
But with 70 seconds on the clock, it was wiped out.
Borna Barisic's ball in was behind the runners in the area but an ingenious flick from Joe Aribo took it to the back post.
And there was the captain right on cue when it counted, though Brazilian keeper Matheus looked more than a little suspect as the right-back slid home.
If his first attempted cross required help to find its man, Barisic's second was faultless as he stood it up perfectly for Roofe to nod home from six yards.
But then French referee Francois Letexier was summoned to the VAR monitor where he discovered the ball had just brushed off the Croatian's left wrist in the build-up.
Not a popular call inside Ibrox.
He signalled for a free-kick, though for half a second Ibrox erupted again when they interpreted that gesture as him pointing to the centre-circle.
All of this inside the first seven minutes. But this magical tournament was only just getting started.
Rangers continued to press forward as Aribo flitted from side to side and spotted an opening on the right.
He slid one across goal and Roofe got a toe to it but could only find the front rows of the Broomloan stand.
Another Barisic delivery caused mayhem as red jerseys piled in to block from Tavernier and then Aribo.
And then it was Tavernier's ball from a corner, flicked on by Aribo, that presented Roofe with a massive chance just yards out but he could only smash the crossbar.
In truth it was a half of total dominance save for a couple of minutes in which Ricardo Horta was able to smash one into the stands. And going into half-time, it was about to get much, much better.
Braga were caught cold as Roofe latched on to a raking long ball from Calvin Bassey and he was sent tumbling in the area by Vitor Tormeida.
Penalty and a red card was the call from Letexier. And when Tavernier steps up from 12 yards the act itself is so often a formality.
He likes to chop and change where he puts his spot-kicks and Matheus flew to his right, giving way as the skipper smashed it central.
The second half started in the same pattern with Braga's only notable early effort an ambitious volley from Ricardo Horta as he tried to catch Allan McGregor off his line. After two decades of experience at this level it was hardly likely.
Then it was Connor Goldson's turn to go close as he smashed the post from a set-piece, though the offside flag would have ruled out any goal-bound effort, before Ramsey's header was wide of the mark from another brilliant Barisic ball.
Ibrox roared again when Roofe dinked over Matheus only for the linesman to judge a knife-edge offside call correctly and rule it out.
Ramsey was then next to bundle the ball home only for a foul on keeper Matheus to chalk that one off.
After so many goals ruled out and missed chances, suddenly there came a sucker punch when Braga won a corner out of nothing – and David Carmo planted a header into the bottom-left corner.
Out of nowhere another 30 minutes beckoned in a second leg that really shouldn't have required it.
But nerves were settled on the 101st minute when Aribo found an opening, slid one across goal and finally Roofe's effort from four yards out counted.
Braga might have thought they'd still have a chance with 10 men. With nine? Not a chance.
Substitute Iuri Medeiros wiped out Leon Balogun on the touchline and, for reasons known only to him, decided to react petulantly enough to the first yellow card that he earned another.
Scott Arfield then climbed off the bench at half time in extra time and missed a complete sitter after Fashion Sakala's deflected effort was parried straight to his feet.
But it mattered not – and here are the talking points from an incredible Europa League night.
Born again
The Croatian was cast out of the starting XI reckoning after a no show in February's Old Firm.
But with Calvin Bassey moved across into centre-back Borna Barisic put in a sensational performance.
He was back to his best with so many excellent chances created down his left side and even the ones that weren't quite on the money were dangerous.
Kent linked up brilliantly and the way Barisic overlapped and looked for opposite full-back Tavernier so many times had echoes of the old Gerrard system.
His song to the tune of Karma Chameleon echoed around Ibrox as they saluted a big game performance from a man some were perhaps too hasty to write off, though an early exit through injury is cause for concern.
Captain fantastic
And so to the other full-back who seems hell-bent on bettering his European performances with every round.
This was a tour de force of every attribute James Tavernier has in his locker, starting with the back post dagger that wiped out Braga's lead inside two minutes.
He then added a collection of set-piece deliveries that should have led to another before converting the penalty that put one blue foot in the semi-final.
Leadership? He had it in spades, at one point dragging away an incensed Barisic from a flashpoint that threatened to boil over and potentially even up the numbers.
Plan comes together... eventually
For all the criticism that comes his way for some domestic performances Van Bronckhorst's nous on the European stage is remarkable.
Ramsey's introduction in the No.10 role was a game changer compared to the away leg while Aribo was given license to flit right and left.
Braga couldn't contain this dual threat without giving Barisic and Tavernier far too much space out wide.
GvB released the handbrake on his full-backs and that dragged back Rodrigo Gomes and Andre Castro to leave the likes of Abel Ruiz completely isolated.
He links a mid-game tinker but plan B wasn't required because plan A worked to perfection, even if it needed another 30 minutes to get the result over the line.
Life without Morelos
There was a moment before kick off when Alfredo Morelos emerged at the mouth of the Ibrox tunnel, crutches in tow.
He gazed out wistfully at the turf on which he's created so many memories in this competition since they first made it to the group stage in 2018 under Steven Gerrard.
Can Rangers can perform without their talisman in the big games? On tonight's evidence, yes.
Roofe will have felt desperately unlucky not to have scored minutes in when his bullet header was ruled out by overexcitable officials in the VAR room.
He then cracked the bar from close in, had a goal ruled out for a marginally offside shoulder and could well have buried the chance that led to the penalty had Tormena not sent him sprawling to the deck and taken the matter out of his hands.
And finally, after 101 minutes of play, he tapped home an effort that DID count.
Let's face it, games like the coming Scottish Cup semi-final and now the Europa League meeting with RB Leipzig would be made far easier had the Colombian been available.
But while his season is over, on the evidence seen on Thursday night that shouldn't mean Rangers' is too.
Make us dream
So read the banner at the front of the Broomloan, accompanied by a gigantic illustration of the Coupe UEFA.
And why not?
The players answered the call and with big tournament favourites Barcelona falling at the hands of Eintracht Frankfurt the competition has been blown wide open.
The famous Ibrox atmosphere produced again as Braga completely wilted, as did Borussia Dortmund and Red Star before them.
It makes the 11 men in blue stand a little taller and RB Leipzig are next up.
