Granderson broke the knuckle getting hit by Cesar Ramos' pitch in the fifth. After later scoring, he was replaced defensively by Ichiro Suzuki in the bottom of the inning.
Granderson missed the first 38 games this season with a broken right forearm, the result of a hit by pitch in his first at-bat at spring training.
"Another bump in the road," Granderson said.
Granderson said at this point he was told surgery will not be required.
"It's sad," Yankees second baseman Robinson Cano said. "It's really tough. He's one of those guys you're going to miss. Hopefully he gets back soon."
David Phelps (3-2), who retired his first 13 batters before James Loney hit a one-out double in the fifth, allowed four runs and six hits in 7 2/3 innings. He left in the eighth after he was hit on the right forearm by a drive up the middle by Ben Zobrist.
X-rays were negative, and Phelps did not rule out making his next start.
"It's a little sore," Phelps said. "See how that feels tomorrow morning will be a bigger thing. I'll be fine."
Gardner made it 5-0 with a two-run homer off Roberto Hernandez (2-5) during the fourth. The Yankees took a 3-0 lead in the second on Overbay's two-run double and an RBI single by Nix.
Tampa Bay slugger Evan Longoria went 0 for 4 and had his career-best 16-game hitting streak end.
Nix walked with the bases loaded, Chris Stewart had a run-scoring single and Cano picked up an RBI when he was hit by Ramos' pitch as the Yankees went ahead 8-0 in the fifth.
Matt Joyce had an RBI double during the Rays' three-run sixth.
Nix tripled and scored to put the Yankees up 9-3 on a wild pitch in the seventh.
After falling behind 3-0 in the third, the Rays intentionally walked Cano with runners on first and second and two outs. The move proved successful when Hernandez got an infield pop fly from Vernon Wells.
Hernandez gave up five runs, six hits and three walks in four innings. He was coming off a start last Saturday against Baltimore where he allowed five runs and eight hits over two innings.
"Pitching won the game for them and not for us," Rays manager Joe Maddon said. "I thought we had some good at-bats, but they did beat us on the mound. We have to pitch better to win."
Tampa Bay's Matt Moore will look Saturday to become the fifth-youngest pitcher since 1916 to start a season 9-0 exclusively as a starter. The 23-year old left-hander will go against Vidal Nuno (1-1).
Maddon said AL Cy Young Award winner David Price, who went on the 15-day disabled list May 16 with a left triceps strain, is getting well. "I don't have any kind of finish line there yet," Maddon said. ... The Yankees are 11-5 since May 8.