Payton Tolle shut down the Tigers over seven innings, allowing just two runs on one hit while striking out eight to earn his first win of the season.
Jarren Duran broke a scoreless tie in the seventh inning, going 3-for-5 with a three-run home run to give Boston a 3-2 lead.
Wilyer Abreu and Marcelo Mayer each added an RBI single in the same inning, extending the Red Sox advantage.
Detroit opened the scoring in the sixth on Matt Vierling's two-run single after an error by third baseman Andruw Monasterio.
The Tigers rallied in the eighth on Dillon Dingler's double, but only managed to tie the game.
Ricky Vanasco (0-1) took the loss, allowing four runs in just two-thirds of an inning.
Red Sox outfielder Roman Anthony left the game during his first at-bat with right wrist discomfort.
Next up: The second game of the series features Brayan Bello (1-4, 9.12 ERA) for Boston against Framber Valdez (2-1, 3.35 ERA) for Detroit.
Aroldis Chapman pitched a scoreless ninth for his seventh save of the season.