After initially being diagnosed with a contusion, Cano was sent to a hospital for X-rays, which were negative. The All-Star second baseman is considered day to day, the team announced. Asked when he expected to return, Cano targeted Friday's home game against Baltimore.
"It's swollen right now, and the last thing you want to do is aggravate it,'' he said.
Cano remained in the game long enough to score on a three-run home run by Alfonso Soriano, the next batter, but did not come out to take the field in the bottom of the inning. He was replaced at second base by Eduardo Nunez.
Cano was hit by an 0-2 fastball clocked at 90 mph that ran in on his hands and appeared to catch him near the pinkie finger. He bent over in obvious pain, and was met by manager Joe Girardi and trainer Stevie Donohue, who worked over him for a long time before the slugger finally went to first base.
Soriano hit the next pitch thrown by Happ and flipped the bat away dramatically as he watched the ball's flight from home plate. It landed three sections up in the left-field seats and gave the Yankees a 4-0 lead before an out had been recorded.
Happ is the pitcher who broke Curtis Granderson's right forearm with a pitch back on Feb. 24, in the first game of spring training. Granderson missed the first 38 games of the season due to the injury.
Cano is the best hitter in the Yankees' lineup, batting .305 with 24 homers and 85 RBIs. He will be a free agent after the season and is expected to draw a lucrative multiyear contract.
Information from The Associated Press contributed to this report.