Verstappen was in the battle for pole throughout but it all fell apart for him in the final session.
His first lap was on course to go fastest until the final sector, when he felt he was impeded by team-mate Sergio Perez at the final corner.
Verstappen complained bitterly about it over the radio to his team, swearing, but in truth it looked like Perez was too far ahead to have cost him the 0.252s by which Leclerc was faster on the first runs.
Hamilton was second fastest after the first laps, just 0.056s off Leclerc.
Verstappen's second lap was measured and he was not fastest of all in either the first or second sectors, and then he ran ever so slightly wide at Turn 19.
His lap put him to the top, beating Leclerc by just 0.005s but his transgression at the penultimate corner cost him dearly and he slipped back to sixth.
Verstappen said: "I knew in 19 it was going to be a close call. I had a little mistake in Turn One, so I had to really push for it in the rest of the lap. I didn't even understeer; I just tried to maximise the corner. I misjudged it by a little bit but it is very fine margins when you are pushing to the limit.
"It is a bit unfortunate but it makes Sunday a bit more fun.
"It is not ideal but I have started further back (and still won). If you have good pace, you will move forwards.
He will still be favourite to win from there, so strong is the Red Bull in the race.
Behind him, Pierre Gasly and Esteban Ocon took seventh and eighth in the Alpine, ahead of the second Red Bull of Sergio Perez and McLaren's Oscar Piastri.
Outside the top 10, the big shock was the performance of the Aston Martins.
The team have a significant upgrade on the car but Fernando Alonso was out at the end of the first session for the first time all year and will line up only 17th. Team-mate Lance Stroll was behind him as usual in 19th.
Alonso said the team had been compromised by problems in the practice session before qualifying, when an overheating problem in Stroll's left front wheel forced the team to pit both cars. Stroll did not get out again, while Alonso's session was truncated.
Alonso said: "We should go back and restart the day completely, starting from the morning. We had a terrible session.
"Lance didn't complete any laps and I did only six or seven quality laps with the new package.
"Too many unknowns in terms of how to operate the package and the new car. We went into qualifying a little bit blind and obviously we see the result."
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