Bourn stole a base in the first inning, then scored, and chased down a fly in center field with two men on for the final out, helping the Indians earn their ninth victory in the past 10 games by beating the Miami Marlins 4-3.
Cleveland went 6 for 6 on stolen-base attempts, and Jason Kipnis' two-run single broke a 2-all tie in a wild seventh inning, when the Marlins narrowly missed three chances to throw out Indians on the bases.
"Our speed tonight helped in a lot of ways," manager Terry Francona said. "We did some good things on the bases, and good thing we did, because the Marlins kept pushing."
Chris Perez survived a harrowing ninth for his 16th save in 18 chances. After pinch hitter Greg Dobbs singled home a run to make it 4-3, the Marlins had runners at second and third with one out, but Perez retired Rob Brantly and Placido Polanco for the victory. Bourn ran down Polanco's fly to end the game.
Steals led to all four runs for the Indians, whose stolen-base total was their highest since 2000.
"It's always fun," Bourn said. "It always adds to the offense. You want to take advantage of the times you can get to the next base."
Kipnis had three hits and drove in three runs.
Miami's Jacob Turner allowed six hits and two runs in six innings. He's winless in his past four starts despite an ERA of 3.47.
McAllister allowed three hits and two runs in 5 1/3 innings.
"I felt good," the right-hander said. "My delivery was where I wanted it, and the ball was coming out good, and I was under control the whole time."
With the score 2-all, the Marlins had Drew Stubbs picked off first base in the seventh inning, but he was safe at second when first baseman Logan Morrison made a wild throw there. Following a walk, Stubbs was barely safe at third on the front end of a double steal, and another walk loaded the bases against Dan Jennings (1-2).
Kipnis then lined a two-run single, with Bourn scoring from second when his slide home just beat a strong throw by center fielder Jake Marisnick.
"Bourny made a real good slide, because that was bang-bang," Francona said.
Miami manager Mike Redmond ran onto the field to complain about the calls at third and home, but TV replays showed the umpires were correct both times.
"A lot of close plays that didn't go our way," Redmond said. "The Indians are trying to be aggressive, and that's their game because they can really run."
Cleveland came into the tied for sixth in the majors in stolen bases. Stubbs had two steals, and Mike Aviles added one.
The Indians stole two bases against Turner, three against Jennings and one against Ryan Webb, all of which made for a rough night for catcher Brantly.
"You can't put that all on the catcher," said Redmond, a former catcher. "They're taking advantage where Jacob is not real quick to the plate. That's something he's been working on, and Webb's not real quick to the plate either."
Miami's first hit came on McAllister's 86th pitch, when Brantly singled to start the sixth. He scored on a double by Christian Yelich, and Ed Lucas followed with an RBI single to make the score 2-all and end McAllister's night. Adeiny Hechavarria lined out to center against Allen with the bases loaded to end the inning.
Bourn led off the game with a single, stole second and came home on a single by Kipnis.
Bourn's speed produced another run in the third. He beat out a grounder to shortstop for an infield hit, stole second, advanced on a wild pitch and continued home when catcher Brantly made an errant throw to third.
"Michael's our spark plug," Kipnis said. "He gets us going and gets us moving in the right direction."
Kipnis tried to score from first when Asdrubal Cabrera's two-out single rolled away from Marisnick, but the rookie outfielder scrambled after the ball, then threw a one-hop strike to the plate for the out.
Miami's Giancarlo Stanton is batting .191 with runners in scoring position. ... The Marlins haven't given up a homer in 11 consecutive games, a team record. ... Indians 1B Mark Reynolds turned 30 Saturday. ... More than a dozen members of the 2003 World Series champion Marlins are expected Sunday to celebrate the team's 10-year anniversary.