Victor Martinez hit a three-run homer to back 2, in the opener of three-game series.
Fister (7-13) registered a career-high 13 strikeouts and surrendered just two runs -- one earned -- on four hits to secure the win.
Jose Valverde tossed a 1-2-3 ninth to remain perfect in 41 save opportunities this season.
Ubaldo Jimenez (2-2) was the hard-luck loser, allowing three runs on just two hits while fanning eight over seven frames.
"I thought both starters were great," said Tigers manager Jim Leyland. "Both starters were really good today and that was key. Our guy was just a little bit better than theirs."
Jimenez did not allow a hit and struck out five through three innings but faltered as light rain began to fall in the fourth. Delmon Young singled with one down and Cabrera walked before Martinez belted his ninth home run of the season.
"I couldn't throw a strike to [Cabrera]," said Jimenez. "The pitch to [Martinez], I was trying to go away and I think it got too much of the plate."
The Indians squandered a bases-loaded, two-out situation in the third when Asdrubal Cabrera went down swinging, but manufactured a run in the fifth. Jason Donald led off with a single and reached second on Fister's wild pickoff attempt. Consecutive groundouts by Lou Marson and Sizemore allowed him to score.
Joe Smith spelled Jimenez in the eighth and allowed the Tigers to load the bases before Cabrera's groundout made it a 4-1 game.
A half-inning later, Fukudome's one-out blast to right-center capped the scoring.
Sizemore returned from the 15-day disabled list. He had surgery to repair a sports hernia on July 21, just four days after departing a game at Baltimore with a right knee contusion...The Indians were 1-for-5 with runners in scoring position.