Verstappen clocked a best lap of one minute and 43.922 seconds in the final minutes of a closely fought session in difficult, gusty conditions on the demanding Baku street circuit to claim his first FP3 top spot of the season.
He wound up 0.099 seconds ahead of Russell, who had topped both practices on Thursday, with Ferrari's seven-time champion Lewis Hamilton third, 0.111 seconds adrift.
World championship leader Kimi Antonelli, who missed half of both Thursday's practice sessions, was 0.252 seconds slower than his teammate Russell.
Verstappen, who posted his time when the two Mercedes had already returned to the garage, still found time to allow his frustration with the car to bubble up over the team radio.
"If, for once, this car wouldn't jump like a kangaroo, my God," he said.
Charles Leclerc, who is seeking his fifth pole in Baku and his first win, was fifth in the second Ferrari, followed by Alpine's Pierre Gasly and the McLaren pair Oscar Piastri and world champion Lando Norris.
Isack Hadjar was ninth for Red Bull and Franco Colapinto 10th in the second Alpine.
The drivers will be back out on the track when qualifying begins at 13:00 BST.
The Grand Prix itself is being run on Saturday, rather than the usual Sunday, because September 27th is a national day of remembrance in Azerbaijan.