The trio finished the fourth round at Le Golf National in Versailles at 11-under 273. Jimenez shot a 67 and his rivals had 68s.
Jimenez had a par on the playoff at No. 18, while Canizares twice hit his ball into the water and Molinari found a fairway bunker with his tee shot and the sand again with his third shot, an approach to the green.
Jimenez earned $613,000 for his 17th European Tour victory and second this season, moving him into the top nine of the qualifying list to automatically make Europe's Ryder Cup team.
The Spaniard had a two-shot lead when he teed off on the 18th, but hit his approach into the water and made double bogey.
"It was quite dramatic in the playoff, but my greater experience probably helped me," said Jimenez, who also won the Dubai Desert Classic this season. "And I had the luck to be able to drive off first.
"I do want to play in the Ryder Cup because this year is probably my last chance at my age. I am taking qualifying week by week, but it would make me very proud to be a member of Colin Montgomerie's team."
Canizares' finish was enough to qualify him for the British Open at St. Andrews July 15-18.
Northern Ireland's Rory McIlroy finished in fourth, one shot outside the playoff, after a 66.
Dutchman Robert-Jan Derksen was tied for the lead with Molinari and Canizares teeing off at the 15th, but he found water three times, eventually taking a 10 at the hole. He finished six shots behind the leaders.
Earlier, Matteo Manassero, having started out in tied for 62nd, shot a 65 to climb 39 places and finish tied for 26th.
The Italian teenager, who won the British Amateur title in 2009, believed the round could be a turning point in his bid to establish himself on the European Tour.
"I know I am able to make cuts," he said. "But I have been having one bad day at each tournament. If I can be more consistent and cut out that bad day then I know I will be OK."