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.
