Civale, who leads the majors with 10 wins, had a 3-2 count on Eric Sogard with two out in the fifth inning when he was visited by manager Terry Francona and a trainer. The trainer examined Civale's right hand before the pitcher was removed.
"The tip of his finger was bothering him, so we didn't mess around with that,'' Francona said. "We just got him out of there.''
Francona said the team wants to have Civale see hand specialist Dr. Thomas Graham as soon as possible.
"Rather than feel sorry for ourselves, we'll just try to figure it out the best we can,'' Francona said.
Civale was 10-2 with a 3.48 ERA heading into the game.
After Civale was removed, Bryan Shaw got four outs for the win. Nick Wittgren worked out of a jam in the seventh before Emmanuel Clase and James Karinchak each pitched an inning, finishing a six-hitter.
"The bullpen came in, and it was just kind of like a regular day for them,'' said Cleveland's Josh Naylor, who hit a two-run homer in the fifth inning. "They just came in and did business.''
Information from The Associated Press was used in this report.