Nio's Cumbrian driver Oliver Turvey led for most of the race from second on the grid, after Felix Rosenqvist stopped in the 15th lap to give up his lead from pole.
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Sebastien Buemi finished third in the Renault after piling the pressure on Turvey in the latter stages after having more power in the final laps, but it was not enough to take second place.
In Formula E, pit stops see the drivers swap cars as their battery-powered cars only last for half of the race.
And previously in Formula E, there was a minimum pit-stop time to ensure the drivers' safety.
But this season the FIA scrapped that rule, allowing Audi and Abt to pull off an exceptionally quick switch.
This is Abt's first ever victory in Formula E, while it was Turvey's and his team's first time on the podium.
Turvey's preparations paid off after he trekked the Andes ahead of one of the highest circuits in the world.
Daniel Abt celebrated after winning his first Formula E race
And while Abt became Formula E's first ever German to win a race, his team-mate and defending champion Lucas di Grassi finally secured his first points of the season after some fine driving saw him score, despite a ten-place grid penalty.
Turvey spoke of his delight to finally step foot on the podium for the first time in Formula E, after first racing in the series in 2015.
He said: "It's amazing, absolutely phenomenal, to get my first podium in Formula E and a first podium for the Nio Team, for the mechanics, the team – I was confident we could do it.
"I have had podiums in every race category I have done, so the fact that Formula E is a really tough series, it means a lot. It's a really competitive field – and to be fighting in a world championship makes me really proud.
"To get my first podium is a great milestone. Last year we got our first pole, first podium now and I hope we can carry this momentum forward."
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