Heaney (6-6) allowed six hits over 5 1/3 scoreless innings. Grant Anderson and Will Smith followed before Aroldis Chapman worked around a two-out single in the ninth for his third save and first since being acquired by the Rangers on June 30
After winning just three of their final 11 games before the All-Star break, Texas has opened the three-game series against Cleveland with back-to-back victories.
Cleveland outhit the Rangers 8-5, but left eight runners on base and was shut out for the ninth time this season
Texas loaded the bases after its first three batters of the game reached base safely against Cleveland starter Gavin Williams (1-2), who was making his fifth career start.
Marcus Semien scored on Adolis Garcia's fielder's choice groundout, but Williams escaped further damage when Josh Jung grounded into an inning-ending double play.
Texas was primed for a big inning in the second after Leody Taveras walked to load the bases with one out. Semien's fly ball was caught in foul territory, but Jonah Heim was able to score on right fielder David Fry's throwing error.
After Corey Seager walked to load the bases, Williams struck out Nathaniel Lowe to end the inning.
Williams settled in and retired nine of his final 10 batters faced. He was lifted after giving up two runs on four hits with four walks and four strikeouts over five innings.
Heaney bounced back with a solid performance after giving up eight runs (seven earned) over three innings against the Washington Nationals in his last outing.
Heaney was lifted in the sixth inning with two runners on after Josh Bell drew a one-out walk. Anderson relieved Heaney and induced an inning-ending double-play grounder from Tyler Freeman to keep the Guardians off the board
Chapman struck out Myles Straw with a runner at first to end the game. He has retired 15 of 17 batters faced in five appearances with the Rangers since being acquired from the Kansas City Royals
--Field Level Media