Audi Sport driver Abt was never headed from pole position over 45 laps of the circuit at the old Tempelhof Airport in the German capital.
Defending champion Lucas di Grassi of Brazil matched his best placing of the season in second place to make it a one-two for Audi, with Vergne winning a thrilling battle with Sebastien Buemi (Renault e.dams) for the final podium spot.
Techeetah's Vergne, who won the previous round in his home city of Paris, has a 40-point lead over Britain's Sam Bird with just three rounds remaining of the supercharged season as his main title rival finished seventh for DS Virgin Racing.
"Extending my title lead is amazing," said Vergne, but the Frenchman admitted he was very much playing second fiddle to the home team drivers ahead of him.
Abt, claiming his second career victory, took the points for pole and setting the fastest lap to delight the home crowds.
His only uncomfortable moment came at the recharging stop as teammate di Grassi closed right up, but he was able to pull away again to win by nearly seven seconds.
"I think I'm out of words, incredible," he said.
"An Audi one-two our best over result, and a home win for me, a super result."
Before the race got underway, the fans in Berlin were treated to the sight of 2016 F1 world champion Rosberg testing the #Gen2 car that will be used next season in Formula E.
It was the first public outing for the car and the first drive on a track in a single seater for Rosberg since clinching his title win for Mercedes.
Rosberg, who has become an investor in Formula E, has said he is not interested in a return to racing, but certainly took the new car through its paces during several laps of the circuit.
His former F1 rival, Australian Mark Webber was among the motorsport celebrities at the Berlin round, along with Britain's David Coulthard.
The next round is in Zurich, Switzerland in three weeks time, before two concluding rounds in New York in July, with Vergne firmly in charge in the title race, Abt moving up to fourth with his fine win.