Justin Rose's side went into Sunday with an 11-4 lead in the race to 13 points, after winning the first two sessions 3.5-1.5 and Saturday afternoon’s foursomes 4-1.
Canter picked up the first point when he closed out a 5 and 4 win over Frenchman Romain Langasque.
When Fleetwood, who was up against another Frenchman in Matthieu Pavon, and Tyrell Hatton, who was facing Thorbjorn Olesen, each moved two up with two to play, securing at least a half-point each, victory was assured.
The impressive Pavon found trouble in a greenside bunker at the 17th and was unable to escape, conceding the match to Fleetwood who was safely on the green and confirming Great Britain and Ireland's win.
"It obviously feels great," said Ryder Cup star Fleetwood, who won all four of his points across the week.
"What an unbelievable group of lads we had this week and an unbelievable captain so just a pleasure to be a part of it. Everybody just played so well."
Hatton went on to beat Olesen while Justin Rose and Paul Waring came out on top against Julien Guerrier and Rasmus Neergaerd-Petersen.