Aitken's previous race win this season came at the Feature Race in Baku, and has since picked up three podium finishes.
The Briton started the race from fourth place but found himself in second place as the cars headed into Turn 3 on the opening lap. Deletraz took the lead off the line from pole-sitter Callum Ilott, who failed to get away well and dropped down to fourth place.
After the restart, Aitken put the pressure on de Vries and reclaimed the second position before chasing after Deletraz.
On lap 17, Aitken had closed up onto the rear of the Swiss driver and made the move stick into Turn 6, which Deletraz couldn't respond to.
De Vries found himself under attack by Callum Ilott in the final laps for the last step on the podium, however the experienced Dutchman wasn't phased as he took third place, the third time in four races he has finished in that spot.
After making ground on championship rival de Vries on Saturday Nicholas Latifi slipped back in the championship on Sunday, as he crossed the line in fifth.
He was involved in a late race battle with Mick Schumacher, which saw the German get airborne at the exit of Stowe after running wide.
Schumacher lost touch with Latifi and found himself being attacked by Guanyu Zhou, while Nobuharu Matsushita closed in, eventually overtaking the Chinese driver around the outside of Turn 1 for seventh.
Zhou took the final point on offer in eighth place while his teammate Luca Ghiotto struggled with his tyres, forcing a late pit stop which put him out of contention for points.
Bahrain International Circuit - Winter testing