Many of the biggest heroes for the Tribe on Sunday afternoon weren't even in the linup to start the game.
With the win, the Tribe share first place with the Tigers in the AL Central.
Reynolds pinch hit in the 10th inning with the bases loaded and two outs, singling home the go-ahead run.
An inning before, Bourn scored the tying run with two outs on a single from Michael Brantley.
The two-out hit gave Tigers closer Jose Valverde his first blown save of the year.
Earlier in the inning, Bourn had led off the inning with a pinch-hit walk, stealing second base.
Valverde managed to retire the next two batters before walking pinch hitter Jason Giambi to keep the inning alive.
The clutch, late-inning runs come in a game in which Tribe starter Zach McAllister struggled early, but kept the Indians in the ball game through six innings. McAllister surrendered just two earned runs, both coming on a Brayan Pena home run in the second inning.
Cleveland tied the game in the third, with Jason Kipnis doubling home a pair of runs with two outs off of Tigers starter Rick Porcello.
From there, the Tigers scored their only other run in the fourth inning, thanks in part to a throwing error from Lonnie Chisenhall which put the leadoff man on second base. The unearned run scored to give Detroit a one-run lead.
The Tribe certainly had their chances to even the score again before the ninth.
Perhaps the best opportunity came in the fifth, when Yan Gomes doubled to leadoff the inning. Brantley and Kipnis both walked to load the bases for the middle of the order.
But Porcello induced a ground ball double play from Asdrubal Cabrera to end the frame.
The same happened in the seventh, as Chisenhall and Gomes both reached to start the inning. This time, it was Tigers reliever Phil Coke who got Brantley to ground into a double play.
Gomes went 3-4 on the day, his first three-hit game of his career. Reynolds' run batted in to give the Tribe the lead was his 32nd of the season, good for fifth in the American League.
On Sunday, Indians pitchers kept the tough middle of the order from doing much damage. Torii Hunter and Miguel Cabrera both hit a pair of singles, but Prince Fielder and Victor Martinez behind them went a combined 0-9 on the day.
None of the first six hitters in the Detroit lineup scored or drove in a run on Sunday, which kept the Tribe in the ballgame just enough to make the late-inning comeback.
The Tribe now returns to Cleveland for a makeup double-header against the New York Yankees on Monday.
AL Central Scores for Sunday, May 12
Cleveland Indians @ Detroit Tigers: 4-3 (10 innings)
New York Yankees @ Kansas City Royals: 4-2
Baltimore Orioles @ Minnesota Twins: 6-0
Los Angeles Angels @ Chicago White Sox: 8:05 p.m. start
AL Central Standings for May 13, 2013
Detroit Tigers:...….…20-15....0.0 games back
Cleveland Indians.......20-15....0.0 games back
Kansas City Royals:...18-16…1.5 games back
Minnesota Twins:...…17-16....2.0 games back
Chicago White Sox:....14-20....5.5 games back
AL Central Schedule for Monday, May 13
New York Yankees @ Cleveland Indians - 12:05 p.m. Game One
New York Yankees @ Cleveland Indians - TBD Game Two
Houston Astros @ Detroit Tigers - 7:08 p.m.
Chicago White Sox @ Minnesota Twins - 8:10 p.m.
Kansas City Royals @ Los Angeles Angels - 10:05 p.m.
All times in eastern standard.
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