The German scored his first podium since returning to the all-electric championship after a year away, and also collected McLaren's first podium since its transition from the Mercedes brand.
Rast managed to pick up the lead having been able to undercut early leader Mitch Evans on their attack mode strategies, but did not put up too much of a fight against the charging Porsche-powered pair of Pascal Wehrlein and Jake Dennis.
This put Rast into a battle with Sam Bird and, although Bird tried a pass down the inside of Turn 18 on lap 33, the Jaguar driver could not stop the car in time and Rast got back ahead.
The three-time DTM champion admitted that he had completely run dry of useable energy at the end of the race, not helped by his spell in the lead where his efficiency was hampered through being unable to use a draft.
"I was settling in third [at the start] and trying to just manage the race. And all of a sudden I was leading, and then all of a sudden I was overconsuming a lot because I had no draft anymore," Rast recounted after the race.
"My energy was dropping and everybody behind had more energy, so it was quite a tricky one, I knew that Pascal and Jake had more energy so I just let them go.
"Then it was a fight to the end with Sam, he had 3% more energy and I was trying to defend for the last six laps.
"He just flew by me at Turn 18, he missed the corner, so I passed back and it was just a fight to the very last corner.
"I ran out of energy just before the last corner and somehow I could creep over the finish line and save third place.
"I'm very happy from where I came, especially from Mexico getting no points and having no pace. And then also Thursday here I had no pace but somehow it started to just ramp up.
"I think we can be very happy with what McLaren has achieved this weekend, today with [Jake Hughes'] pole position and a first podium."
Rast was also pleased with his attack mode strategy, where he opted to use his activations early on and this put him in the right place to slot ahead of Bird once the Briton had used his first dose of 350kW power.
His first attack mode came into play after then-leader Mitch Evans had used his, which allowed Rast to return to the racing line ahead of the New Zealander.
"Actually, I think we were quite good with our attack mode; it's very important that you basically don't lose position," he explained.
"If you lose position and you're in attack mode, and behind someone who's also in attack mode, you lose laptime and you have to pass them back, so you're just burning energy.
"I think two times today I was lucky I got nobody in front of me, so I could use my energy and use the attack mode I had."
