Edwin Encarnacion also connected for Cleveland, which leads second-place Detroit by six games in the AL Central. Encarnacion finished with three RBIs, and Francisco Lindor added a solo shot.
The 23-year-old Bieber (2-0) allowed four hits and walked one in his third major league start. The Tigers only pushed two runners past first base in his 97-pitch gem, stranding them both at third.
Bieber became the third Indians pitcher with six or more strikeouts in his first three big league appearances, joining Herb Score (eight in 1955) and Luis Tiant (three in 1964). Zach McAllister and George Kontos each got three outs to finish the five-hitter for Cleveland's eighth shutout of 2018.
Alonso's fourth career grand slam came off Johnny Barbato as part of a five-run seventh that stretched the lead to 9-0. Alonso and Encarnacion each had two of Cleveland's eight hits in the opener of a three-game series.
Lindor's 17th homer in the third opened the scoring. Encarnacion added a two-run shot in the fourth, and Michael Brantley doubled home Tyler Naquin in the fifth.
Indians third baseman Jose Ramirez reached base for the 29th straight game, walking in his first four plate appearances and scoring twice.
Tigers right-hander Mike Fiers (5-4) surrendered four runs in five innings. He has one win in his last seven starts.
Detroit has dropped three in a row and fell to 2-4 on its eight-game road trip.
The Indians are 5-0 against the Tigers at Progressive Field this season, outscoring them by a 28-5 margin. The first pitch was delayed 1 hour, 41 minutes due to rain.
Tigers LHP Francisco Liriano (3-2, 3.90 ERA) will be activated off the disabled list to take on Indians RHP Trevor Bauer (6-5, 2.50 ERA). Liriano strained his right hamstring against the White Sox on May 26 and will be restricted to 75 to 80 pitches in his return.