With former Yankees gathered for the celebration that featured a brief exhibition game, Brignac did his best Babe Ruth imitation with a two-run homer in the second inning and a three-run blast in the fifth.
"It was a good day, we needed this win today and I was fortunate enough to get some big hits," Brignac told reporters after going 3-for-5.
The 24-year-old Brignac, filling in at second base, had not been playing much lately before Saturday's game after hitting just .219 in June and .176 in July. He had hit two home runs and driven in 25 runs.
"Hits are hard to come by sometimes, especially when you're not playing," he said. "But I tried to keep my same approach no matter what, and just tried to hit the ball hard."
The win brought the second-place Rays (55-35) within two games of American League East-leading New York (57-33) in the contest between the teams with the best records in the majors.
Tampa Bay's 6-foot-9 (2.05m) starter Jeff Niemann gave up four runs in 6-1/3 innings to improve his record to 8-2.
A.J. Burnett yielded four runs before leaving in the third inning due to cuts on his hand sustained when he angrily punched the plexiglass lineup holder attached to a door in the Yankees' clubhouse, frustrated by his performance.
Burnett dropped to 7-8 as the Rays rebounded from Friday's comeback victory by New York.
"I'm not surprised that we bounced back," Rays manager Joe Maddon said.
"We've grown in that regard. It was big that our offense showed up. That's what has really been inconsistent for us this year."
(Writing by Larry Fine, Editing by Greg Stutchbury)