According to the Houston Chronicle, the second-year player suffered a break to his fourth metacarpal and likely will be sidelined four to six weeks. Beverley had 10 points and two assists in 15 minutes against the Pistons before leaving.
The injury is the latest to hit Houston's backcourt.
Leading scorer James Harden sat out Saturday night's game with a sprained right ankle. The All-Star guard is averaging 23.9 points per game, although he was held to only 12 in Friday's 114-81 loss at Indiana.
Jeremy Lin, meanwhile, missed his fourth straight game due to back spasms.
Information from The Associated Press was used in this report.