The accident occurred Saturday night when the right-hander sliced the index finger on his pitching hand on the aluminum foil covering a tray. Super glue was applied, but it did not hold sufficiently for Weaver to pitch deep into a game.
"He cut it on the piece of the topping," Cardinals interim manager Mike Shildt said. "We found out about in the seventh inning. It just made sense long-term to give him the day off.
"It's on his right index finger. He probably could have started, but it just made sense to not push it and have him throw 90 or 100 pitches. We didn't want him to miss another start."
"We want to put this behind [Weaver] and have him move on to the next start and give Tyson Ross an opportunity," Shildt said. "It's worked out for Tyson and for us to have a quality starter, and he's available and is on more than regular rest."
The 24-year-old Weaver is 6-10 with a 4.66 ERA in 23 starts. Ross is 7-9 with a 4.38 ERA.
The Cardinals, who are 5½ games back in the National League Central and 2½ games out in the NL wild-card race, are trying to sweep Kansas City on Sunday.
The Associated Press contributed to this report.