The city-based single-seater series began its fifth season with a first-ever race in Saudi Arabia, which also saw the debut of the Gen2 car that can hit a top speed of 280 kph and will have sufficient battery power to last an entire race.
Previously, drivers have had to stop and change cars.
Pole sitter Da Costa overcame an indifferent start after misjudging his grid spot and had to position his car at an angle pointing towards the barriers.
At the halfway stage, Vergne passed Da Costa to snatch the lead, going around the outside of the Turn 18 right-hander.
Vergne's team mate Andre Lotterer also went past Da Costa into second spot before both DS Techeetah drivers were handed drive-through penalties for technical infringements.
Da Costa retook the lead on lap 24 and, after a safety car period of three full laps, ensured he took the chequered flag for his first Formula E race win since the 2014/15 season.
"It's amazing, it's been tough, it's been long months of work and I am really happy with that," the 27-year-old said.
"We have got some work to do, but a good start and we will work on that and keep it going."
After some thrilling overtaking, Brazilian former Formula One driver Felipe Massa came home 14th in his first ePrix with Venturi.
Belgian Stoffel Vandoorne, who said farewell to McLaren F1 for a seat at HWA Racelab, finished in 17th place.
(Reporting by Hardik Vyas in Bengaluru; editing by Ken Ferris)
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