The ART Grand Prix driver, who started second, took the lead at the halfway mark before crossing the chequered flag ahead of Prema's Hauger.
Hauger’s teammate Olli Caldwell completed the podium, with Vesti’s teammate Alexander Smolyar in fourth.
Caio Collet struggled to get his MP Motorsport car off the grid after stalling, pushing him to the back of the grid, but Hauger managed to get a clean start from pole position.
But Vesti had made up the difference by lap five, taking the lead with DRS after having crept towards Hauger in the lead over the previous laps.
Caldwell overtook MP Motorsport’s Victor Martins on lap six, before his teammate Arthur Leclerc did the same.
Smolyar took second on the seventh lap, while Caldwell and Leclerc passed Jak Crawford (Hitech Grand Prix) up into fifth and sixth respectively.
Smolyar briefly took the lead on lap 11 ahead of Vesti while Hauger took second.
On lap 14, Leclerc and Martins collided, with Leclerc clattering across the gravel before coming to a halt, prompting a safety car, with Trident’s Clement Novalak also taken out.
Caldwell was on the back of Smolyar by lap 21 before managing to take second.
There was contact between Crawford and Jack Doohan (Trident) the following lap, pushing the latter down the standings before both were forced to pit with damage.
Caldwell passed Hauger on the final lap before Hauger managed to take back the position, finishing 1.1s down on race winner Vesti in second.
HWA Racelab’s Matteo Nannini finished fifth behind Smolyar, with Hitech Grand Prix’s Ayuma Iwasa in sixth, having started 19th.
Collet finished seventh, having made up 23 places after being dropped to the back of the grid at the start of the race.
F3 returns for round four at the Hungaroring from 30 July - 1 August.
