Reigning series champion Vergne had sat out Friday's shakedown and first practice due to illness, but climbed to as high as second place in the race, before ultimately coming up just short in an exhilarating battle against Maximilian Gunther.
A rapid start from pole by da Costa contrasted with a somewhat sluggish getaway from second-placed Gunther, but the BMW man saw off Andre Lotterer's challenge into the first corner and set his sights on da Costa.
With Porsche driver Lotterer then slipping behind the Mercedes of Nyck de Vries – only to reclaim third place when the latter was subsequently hit with a drive-through penalty for excessive power regeneration - da Costa and Gunther established an early two-second cushion over the chasing pack.
The gap between the two leaders themselves would oscillate between a few tenths and a second over the opening few laps, but when both moved to activate the first of their two four-minute attack mode periods on lap 11, da Costa made better use of the extra power to put himself two seconds clear of Gunther.
By then, da Costa's teammate Vergne had muscled his way through from 11th on the grid to third, although he soon dropped back behind Lotterer, who had activated his first attack mode later.
Coming into lap 15, with the race approaching its halfway point, da Costa eased off dramatically, allowing Gunther to neutralise the gap and attempt to overtake the DS Techeetah driver around the outside of Turn 11.
Da Costa fought off this attack, but then offered little resistance as Gunther executed a move around the outside of the sweeping Turn 1 left-hander.
With both doing significant energy management, da Costa nearly fell into the clutches of Lotterer as he activated his second attack mode, but having retained the position he then eased past Gunther on the run down to Turn 11.
Gunther then followed suit in activating his final attack mode, but da Costa had established enough of a gap with the use of his allotted extra power to remain ahead once Gunther's four minutes ran out.
At that point, Gunther came under immediate pressure from Vergne – who was last of the frontrunners to activate his second attack mode, having not needed it to re-pass Lotterer for third – and had no answer to the reigning champion when Vergne drove down the inside of Turn 1, in the final seconds of him having extra power.
Yet Gunther now had a usable energy advantage over Vergne, and though the reigning champion threw a very late block on the inside line of Turn 1 in the dying minutes that nearly resulted in a collision, Gunther was unfazed – and was told by BMW that Vergne was fighting “a losing battle”.
The prediction ultimately came true as while Vergne managed to fight off another Gunther attack at Turn 1, this time around the outside, he was powerless to stop Gunther from overtaking him with two corners left.
Their battle had left da Costa some 11 seconds in the clear, while Vergne just narrowly held on to make it two DS Techeetah cars on the podium, finishing two tenths clear of Nissan's Sebastien Buemi.
Venturi's Edoardo Mortara made it four top-eight finishes in five races with fifth place, but finished just eight tenths clear of a spectacular Mitch Evans.
The erstwhile points leader started last after a Jaguar qualifying miscalculation, but broke into the top 10 by lap 24 and made further progress to grab sixth place.
Lucas di Grassi led Audi's efforts in seventh, the highlight of his race coming when he overtook both James Calado (Jaguar) and Jerome d'Ambrosio (Mahindra) in a single Turn 3 move.
From podium contention, Lotterer faded to eighth at the end, with Oliver Rowland (Nissan e.dams) and Sam Bird (Virgin Audi) completing the podium. De Vries recovered to 11th following his drive-through penalty.
Mahindra driver Pascal Wehrlein had his points hopes dashed by a puncture, but posted the fastest lap of the race on the final tour, finishing as the final car on the lead lap in 22nd.
Ma Qing Hua (NIO 333) was a lap down in 23rd, while BMW's Alexander Sims suffered a last-lap heartbreak, dropping out of a points-paying position with an apparent failure.