Bottas ousted Hamilton from top spot on the grid at Imola with his final flying lap, edging his Mercedes team-mate by 0.097s with a time of one minute 13.690secs for the 15th pole of his Formula 1 career and fourth this season.
Verstappen will start third, and that after a nervy session in Q2 when he lost power on his first run, described by team principal Christian Horner as an "electrical issue causing a misfire", believed to be a spark plug.
With two and a half minutes remaining, and following another remarkable effort from the Red Bull mechanics to fix the problem, they were able to send Verstappen back out on a set of mediums that allowed him to secure his place in Q3.
The Dutch driver, though, was half-a-second adrift of Bottas, underlining Mercedes' superiority this season, and will have AlphaTauri's Pierre Gasly alongside him on row two following a stunning session for the Frenchman.
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Renault's Daniel Ricciardo netted fifth, while for Verstappen's under-pressure team-mate Alex Albon, who is in danger of losing his seat, it was another tough qualifying session as he exceeded track limits in Q1 and required a late lap to secure his place in Q2.
On his first run in the second session, Albon spun, flat-spotting his medium tyres, forcing him to undertake his second outing on the softs, and then in Q3, he had another lap deleted on his first outing, adding to his angst, leaving him to eventually line up sixth.
The second part of the qualifying session proved to be a disaster for Racing Point, with Sergio Perez and Lance Stroll lining up 11th and 15th, severely harming the team in its battle for third place in the constructors' championship.
For Russell, it is the eighth time in 13 races this season he has reached Q2, while he has now out-qualified a Williams team-mate for all 34 grands prix in which he has competed, missing out on a place in Q3 by less than three-tenths of a second.
As for Vettel, the four-time F1 champion has now failed to reach the top-10 shoot-out for nine consecutive races, finishing almost four-tenths of a second down on Leclerc.
At the end of an intriguing Q1, and given the perfect conditions for qualifying, it was the usual array of suspects in the bottom five.
Given the positivity leading into the weekend for Alfa Romeo, which confirmed it is to continue in F1 for next season, and with Raikkonen and Giovinazzi signing new deals for 2021, there will have been grave disappointment at such a miserable qualifying.
Raikkonen was one of a handful of drivers, along with Hamilton, Bottas and Albon, who had a lap time deleted in Q1 that would have been good enough to have secured the Finn a place in the second qualifying session.
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