"Just a (bad) day. We need to go home," said Toronto manager John Gibbons, whose team went 3-7 on a 10-day trip that ended Sunday.
Cleveland outscored Toronto 23-5 in the three games.
Kluber (8-3) had season highs in strikeouts and pitches (120) while dominating the Blue Jays' lineup. Kluber struck out every Toronto hitter at least once. After the first inning, 14 of the 20 outs that Kluber recorded were strikeouts.
"Their hitters were aggressive the whole series and we did a pretty good job of using that aggressiveness, not just to get strikeouts but to get quick outs," said Kluber, who is third in the American League in strikeouts despite missing most of May with a lower back strain.
In 10 starts since coming off the disabled list, Kluber is 5-1 with a 1.52 ERA while averaging 13.7 strikeouts per nine innings and holding opposing teams to a .163 batting average.
"He's one of the best in baseball, and that may have been the best I've seen him," Gibbons said.
"He's throwing the ball great, that's for sure," Cleveland manager Terry Francona said. "We rely on him so much, and he knows that. It's nice to put his name in there every fifth day."
Toronto starter J.A. Happ (3-7) gave up seven runs and nine hits with three strikeouts and three walks in six innings.
Toronto's only run came on a homer by Kevin Pillar leading off the third inning.
Kluber allowed five hits and walked one. He has struck out eight or more batters in a club-record 10 consecutive starts.
Cleveland took command of the game early, scoring four runs in the first inning against Happ, who gave up three hits on his first four pitches. Carlos Santana singled and went to second on a bunt single by Francisco Lindor.
Brantley singled home Santana with the first run. One out later, Jose Ramirez drew a walk, loading the bases. Brandon Guyer then drove a double into the left-center field gap, scoring all three runners and giving Cleveland a 4-0 lead.
"It's always nice to score early like that, and separate right away," Kluber said. "It's a boost to score, let alone four runs. It kind of puts the other guy on the ropes."
Gibbons didn't like the view from the ropes.
"After the first inning, I actually thought Happ threw pretty well. But with Kluber on the mound, you know he's not going to cough up many runs," Gibbons said.
Pillar's homer in the third cut it to 4-1, but Cleveland made it 5-1 in the fourth when Giovanny Urshela singled, went to second on a wild pitch and scored on a single by Lindor.
Brantley extended the lead to 7-1 with a two-run homer in the sixth inning. It was his sixth home run of the season but his first since April 30. He went 54 games between home runs.
"I don't get caught up in how many, or when they come," Francona said. "With the quality of Michael's at-bats, the home runs will come. But it's his gap power that is more important than anything."
With the win, the Indians are 24-24 at home, the first time in two months they haven't had a losing record at home. The sweep of Toronto was also big for the Indians, who returned home after losing five of six games to Oakland and San Francisco on their last road trip.
"Hopefully, these three games can jump-start us into playing better and more consistently," Kluber said.
NOTES: Toronto acquired INF Rob Refsnyder from the Yankees in exchange for 1B Ryan McBroom. Refsnyder will report to Triple-A Buffalo. ... Indians 1B Carlos Santana hit in the leadoff spot for the first time since May 13. ... The Blue Jays have a minus-65 run differential in July, their worst month for run differential since minus-61 in June 1979. ... Indians 1B Carlos Santana hit in the leadoff spot Sunday for the first time since May 13. ... SS Francisco Lindor's walk-off home run Saturday night was the Indians' first since last Aug. 19 when OF Tyler Naquin hit an inside-the-park homer, also against Toronto.