The Mexican driver took advantage of a pitstop under the safety car to emerge in the lead after Verstappen had led the early phases of the race from polesitter Charles Leclerc.
Leclerc led away at the start line from Verstappen and, without DRS being made available in the opening series of laps, was able to hold a scant lead over the two-time champion.
But at the end of the third lap, Verstappen breezed past the Ferrari driver thanks to the great speed advantage of his Red Bull with DRS active, and led into the first corner of the fourth lap.
Leclerc quickly fell out of range and was soon easy pickings for Perez, who then immediately began to chase after Verstappen in his efforts to secure a second career victory at the Baku circuit.
Perez had honed to within DRS range of Verstappen, but the race was interrupted to pause the sprint race winner's progress when Nyck de Vries crunched his front-left wheel against the inside wall at Turn 6, and stopped on track.
Verstappen pitted in response, but the emergence of a lap 11 safety car as the marshals tried to clear de Vries' stranded AlphaTauri ensured that his side of the garage had somewhat mistimed the call - the safety car emerging when Verstappen was exiting the pitlane.
This gave Perez and Leclerc the chance to take cheaper pitstops under the safety car, allowing them to emerge from the pitlane in front of Verstappen once they'd collected their fresh hard tyres.
Although Leclerc tried to take a look at Perez on the restart, sticking with the leader, he could not keep Verstappen from blasting past at Turn 3 to assume second place.
But Perez had broken clear of DRS range, crucially ensuring that Verstappen could not employ the powerful rear wing against his own team-mate.
Perez and Verstappen then began to trade blows, and a tug of war over the fastest lap ensued - but the gap began to slowly open in Perez's favour, particularly when Verstappen started to complain of a lack of balance between his differential and engine braking.
By the end of lap 36, Perez was clocking in with laps in the 1m44s, with Verstappen still mired in the 1m45s to help the Guadalajara-born driver's lead grow to 2.5s - which broke the three-second mark two laps later as Perez continued his impressive pace.
Continuing to set the pace, Perez got the gap up to 3.6s but lost 0.6s to Verstappen in trying to lap the twice-stopped Valtteri Bottas' Alfa Romeo, forcing him to bolster his defences in the following lap with another series of quick laps.
The difference between the two was at its zenith at 3.7s, which Perez felt was enough to secure the victory - and began to back off, crossing the line with 2.1s for his second win of 2023.
Verstappen, in search of consolation, went in search of the fastest lap point; although the reigning champion duly delivered the quickest lap, Fernando Alonso swiped the point away provisionally until Verstappen set a 1m44.474.
But Alonso, undeterred, went two-tenths quicker than his fellow two-time champion to add to his points tally while trying to hunt down Leclerc.
Alonso was 0.8s off Leclerc by the end, missing out on a fourth consecutive third place, as the Monegasque managed to shake off the Ferrari's greater tyre degradation to ensure he could convert pole into a podium.
Alonso had earlier made up ground at the restart with an audacious move down the inside of Carlos Sainz at Turn 4, and Sainz finished 23.4s behind his countryman after multiple discussions over strategy were aired on the TV feed.
Lewis Hamilton dispatched Lance Stroll to take sixth, while George Russell - after a good start to the race - fell behind his team-mate and the Aston Martin driver at the restart and struggled to keep tabs with the pair ahead.
The Briton instead elected to call in and bolt on the soft tyres, swiping the fastest lap point away from Alonso.
Lando Norris had spent most of the race sat behind Nico Hulkenberg, who did a long stint on the hard tyre, but eventually broke past the Haas driver to move up into the points - which became ninth when the similarly late-stopping Esteban Ocon called in on the penultimate lap.
This elevated Yuki Tsunoda into the points to ensure AlphaTauri could double his season's points tally - 2.6s clear of Oscar Piastri.
F1 Azerbaijan Grand Prix results:
| Cla | Driver | Chassis | Engine | Gap | Interval | |
|---|---|---|---|---|---|---|
| 1 |  Sergio Perez |
 |
Red Bull | Red Bull | ||
| 2 |  Max Verstappen |
 |
Red Bull | Red Bull | 2.137 | 2.137 |
| 3 |  Charles Leclerc |
 |
Ferrari | Ferrari | 21.217 | 19.080 |
| 4 |  Fernando Alonso |
 |
Aston Martin | Mercedes | 22.024 | 0.807 |
| 5 | Carlos Sainz |
 |
Ferrari | Ferrari | 45.491 | 23.467 |
| 6 |  Lewis Hamilton |
 |
Mercedes | Mercedes | 46.145 | 0.654 |
| 7 |  Lance Stroll |
 |
Aston Martin | Mercedes | 51.617 | 5.472 |
| 8 | George Russell |
 |
Mercedes | Mercedes | 1'14.240 | 22.623 |
| 9 | Lando Norris |
 |
McLaren | Mercedes | 1'20.376 | 6.136 |
| 10 |  Yuki Tsunoda |
 |
AlphaTauri | Red Bull | 1'23.862 | 3.486 |
| 11 |  Oscar Piastri |
 |
McLaren | Mercedes | 1'26.501 | 2.639 |
| 12 |  Alex Albon |
 |
Williams | Mercedes | 1'28.623 | 2.122 |
| 13 |  Kevin Magnussen |
 |
Haas | Ferrari | 1'29.729 | 1.106 |
| 14 |  Pierre Gasly |
 |
Alpine | Renault | 1'31.332 | 1.603 |
| 15 | Esteban Ocon |
 |
Alpine | Renault | 1'37.794 | 6.462 |
| 16 |  Logan Sargeant |
 |
Williams | Mercedes | 1'40.943 | 3.149 |
| 17 |  Nico Hulkenberg |
 |
Haas | Ferrari | ||
| 18 |  Valtteri Bottas |
 |
Alfa Romeo | Ferrari | ||
 Zhou Guanyu |
 |
Alfa Romeo | Ferrari | |||
| Nyck de Vries |
 |
AlphaTauri | Red Bull | |||
| View full results | ||||||
