Phillips (1-0), who was making just his third career start after injuries and illness prompted his call-up from Triple-A Louisville, surrendered just two runs on three hits and a walk while striking out seven for his first big-league win. He gave up solo home runs to Royce Lewis and Alex Kirilloff but retired 12 of the final 13 batters he faced.
Will Benson homered and drove in three runs for the Reds (79-73), who won for the sixth time in eight games. They began the night one half-game behind the Miami Marlins and Chicago Cubs, who were tied for the National League's third and final wild-card spot.
Minnesota starter Joe Ryan (10-10) went five innings, giving up four runs on five hits and three walks. He struck out five.
The Twins (79-72), who lead the American League Central, missed a chance to further lower their magic number. It stood at five going into the game because the Cleveland Guardians lost to the Kansas City Royals earlier Monday.
Kirilloff's ninth home run of the season, a solo shot, trimmed the Twins' deficit to 4-2 in the top of the seventh, but the Reds answered against reliever Dallas Keuchel. Steer's single scored the first run, and Votto's hit went off first baseman Kirilloff's glove and into right field for a 7-2 lead.
The Reds took a 2-0 lead in the second inning on Noelvi Marte's RBI single and Benson's deep sacrifice fly to center, which may have been a home run if not for a catch by Michael A. Taylor near the top of the wall.
Lewis' homer leading off the fourth brought the Twins within 2-1. It was his 15th long ball of the season and third in five games.
The Reds scored two runs in the bottom of the inning when Marte walked with two outs and Benson left no doubt with a 408-foot homer to center.
—Field Level Media