Smoak tied the game with a homer in the ninth inning and knocked in the winning run with a two-out single in the 10th as the Blue Jays beat Indians 2-1 on Tuesday night.
Toronto's Ken Giles (2-2) pitched one inning and earned the win.
The Indians carried a three-hitter into the ninth before Smoak played the role of spoiler, homering with one out off closer Brad Hand. Smoak's 17th homer was his second in two days. He hit a solo shot in Monday's 7-3 loss to Cleveland.
"We faced some tough guys tonight, Bauer and then one of the best closers in the game in Hand," Smoak said.
Hand squandered a save opportunity for the second time in 29 chances.
"Trevor pitched a really good game," Hand said. "It's a tough one to swallow."
Hand, who gave up a run while nailing down the save in Sunday's 5-4 win over Kansas City, was scored on in consecutive appearances for the first time this season.
"I feel like my last few outings, my breaking ball has been real good," Hand said. "I wouldn't say it was bad today, it just wasn't well located."
Adam Cimber got the first two outs in the 10th inning. Olson came on and walked Sogard, who advanced to second on a wild pitch. Sogard moved to third on an infield single by Freddy Galvis and scored when Smoak singled off the glove of a diving Jose Ramirez at third base.
The Blue Jays beat the Indians for the first time in six games this season. Cleveland is 13-4 in July.
Bauer made the start despite feeling ill earlier in the day. He allowed three hits in 7 2/3 innings, walked three and struck out nine. The right-hander didn't allow a hit until Vladimir Guerrero Jr. doubled to start the fifth.
"I don't feel too good now, but I was fine during the game," Bauer said.
Indians manager Terry Francona saluted Bauer for taking his turn despite the illness.
"I thought he was terrific," Francona said. "He gave everything he had. He always does."
Bauer pitched seven no-hit innings against Toronto on April 4, a game the Indians won 4-1.
"He's tough," Smoak said. "We didn't get much done off of him but our pitchers kept it close enough for us to win the game at the end."
Blue Jays starter Aaron Sanchez allowed one run and five hits in five innings. He has not won since April 27.
"A step in the right direction for sure," Sanchez said.
The right-hander entered the game with a major league-leading 59 walks. He didn't allow a base on balls for the first time in 61 starts, ending the longest active streak in the majors.
Indians: OF Tyler Naquin (right hamstring) was held out of the starting lineup.
Blue Jays: RHP Clay Buchholz (elbow) resumed throwing bullpen sessions last week and is scheduled to throw two more this week. ... OF Dalton Pompey (concussion) was activated off the 60-day injured list and designated for assignment. ... RHP Justin Shafer was able to play catch on the field before the game. Shafer left Monday's game because of a sore elbow.
ON THE MEND
The 3-year-old boy hit by a foul ball in Cleveland last weekend is showing no signs of serious injury, the Indians said Tuesday. The boy, identified as Henry, was hit Sunday by a line drive off the bat of Lindor. In a statement, the Indians said Henry was released from hospital Sunday evening.
Bauer has thrown a major league-leading 2,594 pitches this season. Houston's Gerrit Cole (2,261) is second.
Smoak has 20 home runs in the ninth inning or later over the past three seasons, more than any other player in the majors. San Diego's Hunter Renfroe (13) is second.
Indians: RHP Shane Bieber (9-3, 3.69) is 13-2 with a 3.38 ERA in 21 career appearances on the road.
Blue Jays: RHP Marcus Stroman (6-10, 3.06) pitched seven shutout innings to beat Detroit in his previous start.
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