The Monegasque has won 10 of his 12 meetings with top-50 ranked players this year, with world number 31 Norrie the latest to fall.
Norrie dropped just two points on serve in the opening set but failed to capitalise on his one break point opportunity and eventually came up short in the tie-break.
The Briton was booed by the crowd in the French capital when he tried to get them on board by raising his arms in the second set.
He went on to pass up a further two break points at 2-2 and then dropped serve in the following game.
With momentum on his side and the finishing line in sight, Vacherot saved another two break points to lead 5-3 before cupping his hand to his ear and waving his racquet to the crowd.
He wrapped up victory on his second match point when Norrie hooked a forehand into the base of the net.
World number two Jannik Sinner ensured he became a quarter-finalist in every ATP Masters 1000 event this season with a 7-5 6-1 win over Argentina’s Francisco Cerundolo and will meet American Ben Shelton in the next round.
Third seed Alexander Zverev kept his title defence alive with a 6-2 6-4 victory over Alejandro Davidovich Fokina and will face Daniil Medvedev in the quarter-final after the 11th seed defeated Lorenzo Sonego.
Davidovich Fokina needed a medical timeout in the second set after injuring his left leg, before his German opponent sealed his success.
“He is an amazing player, he is in a good rhythm so I was very happy with the level,” said Zverev. “I thought I played well from the first moment to the last and he is very dangerous so I’m very pleased with the straight-set win.”
