Abt, who started from fifth on the grid, worked his way through the field in the 47 laps of the Autodromo Hermanos Rodriquez to take the checkered flag in his Audi Sport entry, the first win for a German in the series.
Britain's Oliver Turvey (NIO Formula E) finished second, his maiden podium position, with former world champion Sebastian Buemi of Switzerland in third place.
Series leader Jean-Eric Vergne (Techeetah) finished fifth but has extended his overall advantage in the title race to 12 points.
Vergne's closest challenger, Felix Rosenqvist of Sweden, started from pole position for Mahindra Racing, but a technical problem ended his hopes while leading in the early stages.
Vergne, who took the previous round in Chile, was running as high as third until falling victim to a fine overtaking move by Buemi and later lost a further position to Nelson Piquet Jr. (Panasonic Jaguar Racing).
Rosenqvist's misfortune left young hopeful Turvey in the lead, but Abt closed up ahead of the round of pit stops and took the lead at that point before pulling over six seconds clear of his nearest challengers by the finish.
"What a fantastic race," said Abt, who has moved up to sixth in the overall standings.
"I hope we can keep this up, the car was fantastic and my guys did an incredible pit stop," he added.
Abt, who has raced in the series since the inaugural 2014-15 season, was denied victory in the second round of the current season in Hong Kong back in December due to a technical infringement, with Rosenqvist elevated to first place.
Turvey, who held off a charging Buemi at the end, was also delighted by his effort.
"We have been working so hard and showed we have good pace," he said.
Defending Formula E champion Lucas di Grassi, who won the inaugural E-Prix in Mexico City, took his first points of a disappointing season by finishing ninth, but has an uphill battle to retain his title.
Round six of the 12-race championship takes place in Punta del Este in Uruguay in a fortnight.