The rookie, who had homered twice Sunday, lined a pitch from Cleveland reliever Zach McAllister (3-4) into the right field bleachers to give Chicago its fifth straight win and 21st victory in 26 games.
"You want to be in a spot where you're pitching against a guy who's a really good pitcher and he's throwing the ball well," Kluber said. "You kind of go toe-to-toe with him and both give our team chances."
The Indians didn't cash in, however.
"It would be a lot more enjoyment, if we were up like 10-0," Kluber said.
Kluber, the 2014 AL Cy Young winner flirted with a no-hitter for the third time in his last four starts, then allowed just four hits while striking out in 11 in 7 2/3 innings. He retired the first 16 Cubs hitters before back-up catcher David Ross lined a single to center with one out in the sixth.
"He was good. He and Lester both," Indians manager Terry Francona said. "There was absolutely zero wiggle room and he pitched about as good as you can. He executed pitches all day."
But then Bryant became the only player who came close to taking advantage of a steady wind to right with his 20th home run.
"On a day when the wind's blowing out, it wasn't a factor until the last hitter of the game," Francona said.
Bryant said he was ready for his chance.
"It was a great pitchers' duel," Bryant said. "I thought Jon had enough to pull it out but sometimes you've got to give credit to the other team.
"They came through and I knew I was going to get an at-bat there and I made the most of it."
Backed by four double plays, Lester rebounded from his shortest start of the season to pitch his longest since joining the Cubs. He was relieved by Hector Rondon (5-2) with two outs in the ninth after scattering six hits and striking out six.
Lester nearly matched Kluber through the first seven, facing only 22 Cleveland hitters.
Both starters rebounded from shaky previous outings after being on rolls.
Lester had won four straight decisions in five starts before lasting only 2.2 innings against Detroit last Wednesday when he gave up seven runs on seven hits in a 15-8 loss. In his previous five outings, the lefty permitted eight runs on 27 hits and struck out 40.
Indians: All-Star 2B Jason Kipnis was back in the leadoff spot after being held out of the starting lineup for rest on Sunday at New York. Kipnis pinch-hit in that contest.
Cubs: OF Jorge Soler (left oblique stain) and RHP Jason Motte (right shoulder strain) were placed on the 15-day disabled list before the game. Motte won't throw for seven to 10 says. Soler could be out for up to a month.
IF Addison Russell got the day off to be with his fiancee, who is expecting their first child.
Cubs: Right-handed ace Jake Arrieta (15-6, 2.30 ERA), who has won four straight and is 9-1 in his last 12 starts, faces the Giants' Matt Cain (2-3, 5.66 ERA) as the Cubs open a pivotal three-game series in San Francisco.