The four-time world champion clocked a best lap in one minute and 30.029 seconds to grab the 75th pole position of his career ahead of Bottas, giving himself a flawless chance to regain the initiative in the title race in Sunday's race.
"This is a great day", said reigning champion Hamilton.
"I think the vehicle is quick, but I wasn't able to be quick", he said.
Hamilton is also hoping that the race will be more entertaining for the fans at Le Castellet than some of the more recent events which have been criticised as being too boring.
The German's comeback was impressive but fifth would have been as much as he might have expected, while Verstappen put in another very solid performance to take his second podium in a row and his third in four races.
A round-up of the main talking points from the French Grand Prix, where Lewis Hamilton moved back into the lead of the championship with a comfortable win at Paul Ricard. "Then I ended up with nowhere to go".
Mercedes boss Toto Wolff said: "It was a misjudgement from Sebastian because it ruined his race as well".
Hamilton and Vettel now have three wins apiece from eight races in 2018, as they each battle to become only the third driver in history to clinch five F1 drivers' world titles.
"I haven't thought about [the championship lead], but that's where I want to be".
On seeing a video recording of the incident, he said: "He took him right out -- ah, man, that's insane".
Grosjean finished 11 seconds outside of the points, five of those from his time penalty, with McLaren's Stoffel Vandoorne struggling in 12th.
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"I only got a few laps in, then we had a water leak which we're now investigating".
Raikkonen himself did little to dampen the speculation on track, lapping over a second shy of Hamilton and in excess of half a second behind Vettel.
Vettel believes he could have challenged Hamilton for the win, but he said the nature of his race following the incident made it hard to judge what was possible.
Renault's Nico Hulkenberg took ninth place with Monaco rookie Charles Leclerc a fine 10th for Sauber and denying Haas's French driver Romain Grosjean his first point of the season. Hamilton had not been troubled at the front of the race, he managed and maintained the gap to Verstappen to ensure a clinical finish.
Verstappen, who has delivered two successful and composed races since an emotional outburst at reporters in Canada earlier in June, relished his podium finish.
"I tried to follow Lewis, but obviously he was controlling the pace". I enjoyed the race. I've been struggling to get ideal laps this week -- I was catching up rung by rung, but a one-two here is great.
But there his progress stopped and he was able to make a late pit stop for fresh tyres, serving his penalty at the same time, without losing his position.
"But we won't really know its full potential until tomorrow when everyone gets to turn their engines up".
His compatriot Bottas was typically matter-of-fact about his crash with Vettel.
"We were side by side and I went for the outside, left enough room for him, but he went wide and he hit me".
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