The Prema driver lined up sixth, making up two places on the first two laps before closing on the leading trio of Ollie Bearman, Victor Martins and Jack Doohan.
He pitted alongside Doohan, emerging in front of the Virtuosi driver to secure third.
Vesti waited patiently while Bearman and Martins tussled for position ahead, before both rookies spun out moments apart around the midway mark, allowing the Mercedes junior to eventually take the lead once those who started on medium rubber pitted.
He held on to cross the line 3.9s clear of Doohan, who claimed his first podium of the season, with Jehan Daruvala taking his second third-place finish of the weekend for MP Motorsport.
A poor start for Martins saw Bearman take the lead off the line, with Doohan looking to claim second before settling into third behind the ART driver.
There was a virtual safety car on lap two after Brad Benavides and Amaury Cordeel made contact, the PHM driver’s race ended with suspension damage while Cordeel pitted with front wing damage.
Bearman had built a good gap out front when the VSC ended the following lap, with Martins, Doohan and Vesti close behind him.
Oliver Bearman, Prema Racing leads Victor Martins, ART Grand Prix
Photo by: Mark Sutton / Motorsport Images
Vesti and Doohan were the first of the leading quartet to pit on lap six of 28, with Vesti emerging in front into 12th position.
Bearman, Martins and Daruvala pitted the following tour, with the Briton pipping the reigning Formula 3 champion to the pit exit for eighth place.
DAMS’ Arthur Leclerc led those who were yet to pit, with Enzo Fittipaldi (Rodin Carlin) and Hitech rookie Isack Hadjar in second and third.
By lap 10, Martins had caught Bearman, taking the lead of those who had stopped off the road before handing the position back.
He was not behind for long, though, snatching sixth on the start-finish straight with the help of DRS as Bearman was forced wide out of Turn 2, allowing Vesti through into seventh as Doohan closed in from behind.
A chaotic lap 17 saw Bearman spin at Turn 22, dropping from eighth to 11th behind Ayumu Iwasa.
Moments later, Martins also spun at Turn 5, losing his engine and ending his race, prompting another VSC.
Vesti inherited sixth position, ahead of Doohan, and eventually moved into the lead on lap 24 as Leclerc, Hadjar and Trident’s Roman Stanek pitted.
Iwasa finished in fourth, with Dennis Hauger in fifth for MP Motorsport and Verschoor in sixth.
Fittipaldi took seventh, with Leclerc in eighth, Hadjar in ninth, and Bearman claiming the final points paying position after his spin from what could have been a podium position.
Theo Pourchaire, who was handed a five-place grid penalty after colliding with Bearman in Saturday’s sprint race, lined up eighth, but endured another difficult race to finish in 13th.
F2 Saudi Arabia feature race result
