Bruce, acquired in a trade from the New York Mets on Wednesday night, drove in the first run with a two-out fly ball that dropped for a double between left fielder Corey Dickerson and third baseman Trevor Plouffe in the first inning.
After leading off the third with a double, Francisco Lindor scored on Chris Archer's two-out wild pitch to make it 2-0.
Bruce drove in another run with a single in the fifth after Jose Ramirez's leadoff double.
Clevinger (6-4) won for the first time in four starts since the All-Star break, giving up four hits and one walk in seven innings.
Cody Allen pitched the ninth for his 20th save.
It was 10th time the Rays were shut out this season, most in the majors.
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