Michael Brantley added an RBI groundout in the first inning and a run-scoring single in the fifth.
John Axford pitched the final 1 2-3 innings for his 10th save.
Cincinnati manager Bryan Price praised Kluber's performance. "As advertised," Price said. "I know he's a tremendous pitcher, we all knew that coming in. He did the things that we knew about. He's very good."
Center fielder Michael Bourn will begin a minor league rehab assignment at Double-A Akron on Tuesday. He hasn't played since July 5 because of a recurring left hamstring injury.
Reds: Second baseman Brandon Phillips is traveling with the team as he continues to rehab from a torn ligament in his left thumb. Price said Phillips, who hasn't played since July 9, still can't swing a bat.
Indians: Designated hitter Nick Swisher is optimistic his strained right wrist won't sideline him for more than a couple of days. "I've been through stuff like this before," he said. "It's just a little tight." He injured the wrist in his last at-bat Sunday against Texas' Yu Darvish.
Reds: All-Star right-hander Johnny Cueto (12-6) will start Tuesday. He leads the NL in innings pitched (162 2-3) and opponents' batting average (.183) and is second in ERA (2.05) and strikeouts (186). Cueto is 3-0 in five career starts against Cleveland.
Indians: Right-hander Josh Tomlin (5-7) will be called up from Triple-A Columbus to start Tuesday. He was sent to the minors on July 26. Tomlin hasn't won since pitching a one-hitter on June 28. He's 1-1 in three career appearances against Cincinnati.