The Briton lapped the Baku city circuit with a best time of one minute and 42.514 seconds, with Ferrari's Charles Leclerc 0.013 slower.
Leclerc has been on pole in Baku for the past three seasons and was fastest in second Friday practice.
Russell's lap was something of a surprise, although the track was getting faster and he set his time later than Leclerc, after he had complained that "the ride feels terrible down the straight under braking".
McLaren's Lando Norris and Oscar Piastri were third and fourth fastest, ahead of the Red Bull of championship leader Max Verstappen who has a 62-point lead over Norris in the standings with eight rounds remaining.
Verstappen was fastest in Friday's first practice, meaning three different teams had topped the three sessions in Baku before Saturday's qualifying.
Ferrari's Carlos Sainz was sixth and Red Bull's Sergio Perez seventh on a grey day with some spots of rain.
The two Williams of Alex Albon and rookie Franco Colapinto were eighth and ninth and Mercedes' Lewis Hamilton rounded out the top 10.
The red flags came out after Esteban Ocon's Alpine lost power and stopped before anyone had set a timed lap and then British rookie Oliver Bearman, a stand-in at Haas for the suspended Kevin Magnussen, hit the wall.
Bearman realised too late that he was not going to make the corner at turn one and tried to steer clear down the escape road but swiped the barrier with his front left.
"Oh, I'm such an idiot," he said over the radio.