The Briton was an unlikely contender to top qualifying after the struggles for the Mercedes W13 all season, not least during both dry and wet running at the Hungaroring so far this weekend.
But despite not taking a fastest sector on his critical lap, his combined runs sailed him to the top over Carlos Sainz, who had the legs on his teammate Charles Leclerc.
But it was a session to forget for Red Bull, with Verstappen only poised to line up 10th as Sergio Perez was eliminated in Q2.
The day-long downpour that was widely predicted held off to allow qualifying to take place as scheduled on Saturday afternoon in Budapest.
It was Sainz who ended the first part of the 12-minute Q3 run on top, his first flying lap placing him half a second clear of Russell after Leclerc had messed up straight away with a snap in his F1-75 at Turn 1 as he struggled with the tyres.
With Verstappen combatting understeer to run off track at Turn 2, he ended the first run-out only seventh fastest.
The 10 fastest cars then emerged with 3m40s to run, but Verstappen was soon complaining of no power aboard his RB18 - cycling through sensor modes not enough to remedy the problem.
With the defending champion, who leads Leclerc in the standings by 63 points, out of the picture, Sainz cycled to the top with the fastest first sector of anyone (pipping Nicholas Latifi).
That gave the British GP polesitter the bragging rights initially as Leclerc slotted in 0.15s adrift, but then three personal best sectors at the death for Russell gave him an unlikely pole.
His 1m17.377s effort pipping Sainz's 1m17.421s best, while Leclerc managed to keep ahead of the McLaren of Lando Norris - the MCL36 impressing in dry qualifying and race simulations during FP2 on Friday.
Esteban Ocon managed to reverse the recent run of form to pip Fernando Alonso in the intra-team Alpine scrap, while Lewis Hamilton's DRS failed to activate on his final lap to knock him to seventh.
Valtteri Bottas claimed eighth ahead of Daniel Ricciardo, as Verstappen had to make do with a 1m18.823s that left him 10ths and 1.5s adrift of the Mercedes benchmark.
Perez was the major casualty from Q2 in a strange session for the driver, his Red Bull being knocked out by a slender 0.05s courtesy of the late improvements from Ocon and Bottas.
The Mexican complained that Kevin Magnussen's line through the Turn 3 left-hander had ultimately cost him a place inside the top 10.
Perez had his first lap in Q2 scrubbed off for exceeding track limits at Turn 5, but it seemed a marginal call as replays showed not all four wheels crossed fully over the white lines.
Seven minutes after the fact, the lap was therefore reinstated to put him eighth. The reports of spots of rain plus a dip in track temperatures might have kept him safe enough.
But as the showers held off, the late climbs from the Alfa Romeo and Alpine dropped Perez to 11th ahead of Zhou Guanyu and Magnussen.
The Dane aborted his final lap after missing his personal bests in sectors ones and two.
Lance Stroll, meanwhile, will start 14th ahead of the second Haas of Mick Schumacher.
Perez had already been at risk in 14th in Q1 as the chequered flag came out, but he was given respite when Pierre Gasly had his improved lap time deleted for exceeding track limits at Turn 5.
That dropped the AlphaTauri to 19th, with his teammate Yuki Tsunoda also eliminated in the first part of qualifying – the Japanese driver similarly crossing the white lines earlier in the session at the same protracted right-hand bend to have to start all over again.
With Tsunoda's final, compliant effort not enough to progress as he landed 16th, Stroll had secured his slot in Q2 by 0.35s. Meanwhile, Alex Albon ran for 17th ahead of the hastily repaired Aston Martin of Sebastian Vettel.
In the two-hour break between sessions, the German had helped his crew repair the AMR22 machine following his shunt into the barriers late in FP3 that resulted in a brief red flag.
The retiring four-time champion, who appeared to struggle for balance through the final corner, ultimately landed only 18th.
Latifi, the unlikely pacesetter in a slippery final practice, might have progressed into the Q2 fight when yet again he mastered the first sector to set a purple run late on.
He combined this with a personal best sector two only to suffer a spike of oversteer in the final right-hander to cost him 0.7s and leave him last in qualifying - much to his frustration over team radio.
| Cla | Driver | Chassis | Engine | Time | Gap | |
|---|---|---|---|---|---|---|
| 1 |  George Russell |  | Mercedes | Mercedes | 1'17.377 | |
| 2 |  Carlos Sainz |  | Ferrari | Ferrari | 1'17.421 | 0.044 |
| 3 |  Charles Leclerc |  | Ferrari | Ferrari | 1'17.567 | 0.190 |
| 4 | Lando Norris |  | McLaren | Mercedes | 1'17.769 | 0.392 |
| 5 |  Esteban Ocon |  | Alpine | Renault | 1'18.018 | 0.641 |
| 6 | Fernando Alonso |  | Alpine | Renault | 1'18.078 | 0.701 |
| 7 | Lewis Hamilton |  | Mercedes | Mercedes | 1'18.142 | 0.765 |
| 8 |  Valtteri Bottas |  | Alfa Romeo | Ferrari | 1'18.157 | 0.780 |
| 9 |  Daniel Ricciardo | | McLaren | Mercedes | 1'18.379 | 1.002 |
| 10 |  Max Verstappen |  | Red Bull | Red Bull | 1'18.823 | 1.446 |
| 11 |  Sergio Perez |  | Red Bull | Red Bull | 1'18.516 | 1.139 |
| 12 |  Zhou Guanyu |  | Alfa Romeo | Ferrari | 1'18.573 | 1.196 |
| 13 |  Kevin Magnussen |  | Haas | Ferrari | 1'18.825 | 1.448 |
| 14 |  Lance Stroll |  | Aston Martin | Mercedes | 1'19.137 | 1.760 |
| 15 |  Mick Schumacher |  | Haas | Ferrari | 1'19.202 | 1.825 |
| 16 |  Yuki Tsunoda |  | AlphaTauri | Red Bull | 1'19.240 | 1.863 |
| 17 |  Alex Albon |  | Williams | Mercedes | 1'19.256 | 1.879 |
| 18 | Sebastian Vettel |  | Aston Martin | Mercedes | 1'19.273 | 1.896 |
| 19 | Pierre Gasly |  | AlphaTauri | Red Bull | 1'19.527 | 2.150 |
| 20 | Nicholas Latifi |  | Williams | Mercedes | 1'19.570 | 2.193 |
| View full results | ||||||
