Manager Terry Francona was forced to turn the game over to the bullpen.
It turned out that the game was in good hands.
After six relievers held the Blue Jays to two runs, the Indians will aim for a sweep of the best-of-seven set on Tuesday at Rogers Centre. Game 1 winner Corey Kluber will start for Cleveland against Toronto's Aaron Sanchez.
"Tito (Francona) did a masterful job of running that bullpen today," Blue Jays manager John Gibbons said. "They shut us down. I'm not going to get into who did what, this or that. But they did a great job going through a number of guys and did a good job of shutting us down."
Bauer left the game after with two outs in the first inning following his second walk, as his previously injured right pinky finger began bleeding while he was on the mound.
"That wasn't the way we drew it up," Francona said. "But our bullpen, that's one of the most amazing jobs I've ever seen. ... If anybody has a hiccup, we probably lose. And (we) made all the pitches against some really good hitters."
Napoli said of Cleveland's relievers, "For them to come in and pitch the way they did, it's pretty unbelievable. It's the reason we were able to win that game, especially a lot of those guys, they haven't pitched in a long time. We had a simulated game before we left to come (to Toronto). For them to step in, I think you have to give a lot of credit to Roberto Perez behind the plate being able to get those guys to execute pitches."
Toronto starter Marcus Stroman (0-1) allowed three hits, including two home runs, three walks and four runs in 5 1/3 innings.
Bauer was trying to pitch after cutting his finger while playing with a drone. The injury required several stitches and prevented him from starting Game 2.
"The first thing I saw was blood on the rubber," Francona said. "I figured that it wasn't a real good sign that things were going well. It was bleeding pretty good. The weird thing, it was only one little spot that opened, but, man it was coming out a lot."
Kipnis led off the sixth inning with his second home run of the postseason, a liner to right on a 2-2 Stroman fastball, that gave Cleveland a 3-2 lead.
Joe Biagini replaced Stroman after a one-out walk to Napoli in the sixth. Napoli took second on a wild pitch and scored on a single by Jose Ramirez, and Cleveland led 4-2.
Miller pitched around a leadoff single from pinch hitter Dioner Navarro in the ninth to earn the save. He had three strikeouts in 1 1/3 innings, giving him 13 K's over five innings in the series.
Cleveland took a 1-0 lead in the first on a two-out RBI double by Napoli. The hit scored Carlos Santana, who led off with a walk.
Bauer came out after he gave up his second walk to his fourth batter, Troy Tulowitzki. Francona and a trainer checked out Bauer before Dan Otero was summoned from the bullpen.
The Blue Jays tied the game on the first postseason home run by Saunders, a drive to left on a 1-1 Otero changeup while leading off the bottom of the second.
Jeff Manship replaced Otero for the third and allowed one hit in 1 1/3 innings.
Napoli led off the fourth inning with his first homer of the postseason on a 0-1 fastball to give Cleveland a 2-1 lead.
Zach McAllister allowed a leadoff triple to Carrera in the bottom of the fifth. Carrera scored the tying run on a groundout to shortstop by Ryan Goins.
"We've won four games in a row plenty of times," said Toronto reliever Jason Grilli, who allowed one hit in two-thirds of an inning. "I'm not going to come up and be a soap box, rah-rah thing. I don't even know how to answer some of these questions, honestly. I don't know what I can really say at this point other than we're going to come out and play tomorrow, play like we have, play for the city, like we have."
NOTES: Toronto RF Jose Bautista was the leadoff hitter Monday in a shuffled lineup with SS Troy Tulowitzki replacing him in the No. 4 spot. ... Blue Jays 2B Devon Travis was replaced on the ALCS roster by 1B Justin Smoak. Travis aggravated a right knee injury during Game 2. An MRI exam revealed a bone bruise with a small flap of cartilage in the joint. He likely will require arthroscopic surgery and is expected to be ready for spring training. ... Cleveland became the second team to take a 2-0 postseason series lead by scoring two or fewer runs in each game, winning 2-0 and 2-1 at Progressive Field. The 1950 New York Yankees defeated the Philadelphia Phillies 1-0 and 2-1 on their way to a four-game World Series sweep. .... Toronto RHP Aaron Sanchez (0-0, 9.53 ERA in postseason) will face Cleveland RHP Corey Kluber (2-0, 0.00) Game 4 on Tuesday night.