The Estonian driver, who said after the lap that he “thought he completely missed it [pole],” set a 1m53.762s to top the timesheet after a red flag disrupted running.
Vips took advantage of clear air in the middle sector, making up four-tenths on his previous lap to claim the front spot for Sunday’s feature race, having been sitting in sixth.
As a result, he pipped 2021 polesitter Liam Lawson, who had been fourth before the final runs, by 0.163s.
Dennis Hauger took his personal best qualifying result with a third for Prema, but the rookie is under investigation for an unsafe release in front of Roy Nissany as the field left the pits for the final runs.
Jake Hughes was first out the pits for Van Amersfoort Racing, putting in a 1m56.556s as the first flying lap on the board.
Ayumu Iwasa topped the timesheet on his first attempt for DAMS, setting a 1m54.483s straight out the box, just two-tenths off Lawson’s 2021 pole lap, a 1m54.2s.
With the first flying laps complete, Iwasa led from Hauger, who was 0.6s off the pace, with championship leader Felipe Drugovich in third for MP Motorsport, 0.1s behind the rookie.
The yellow flags were waved in sector one 10 minutes into the session after Ralph Boschung, returning from injury, went deep into Turn 3 in his Campos Racing-run car.
Marcus Armstrong was next to set a quick lap for Hitech, going second quickest with a 1m54.487s, just 0.004s off Iwasa, while Drugovich remained in third having improved on his earlier attempt.
The field pitted at the halfway mark, emerging with 11 minutes left on the clock for their second runs.
But the red flag was thrown with just under six minutes remaining after Iwasa ran into the wall on the exit of Turn 3, his front right wheel colliding with the barriers.
Once running resumed, it was a mad dash to the line for the field, with Hauger released into the path of Nissany as the pack headed out on track.
Richard Verschoor briefly claimed the top spot, with a 1m54.029s taking him to the top of the timesheet having been sat in 11th place.
But as the times tumbled, he was knocked down to sixth place, with Vips taking the top spot.
Hauger briefly slotted into second, having taken advantage of the tow from Calan Williams, before he was demoted by Lawson, with Armstrong taking fourth place.
Drugovich will line up fifth, with Carlin’s Logan Sargeant in seventh.
Hughes will start on pole for Saturday’s sprint race, with Frederik Vesti lining up second for ART Grand Prix and Prema’s Jehan Daruvala in third.
FIA F2 Baku qualifying results
