''We got a couple of big hits and we were able to piece it together,'' Indians manager Terry Francona said.
Scott Atchison (4-0) pitched 1 2-3 innings and worked out of a two-on, no-out jam in the sixth.
Cody Allen worked a shaky ninth for his 14th save. J.P. Arencibia drew a leadoff walk and Leonys Martin singled, but Allen regrouped. Pinch-hitter Jim Adduci struck out and Rougned Odor fouled out before Daniel Robertson struck out to end the 24-pitch inning.
''Luckily, I was able to make some pitches to get out of it,'' Allen said. ''You're going to get in jams. You're facing very good hitters who are going to beat you. It's just part of being a closer.''
The win moved the Indians (55-55) back to .500 while the Rangers have dropped 18 of 21 away from home and are the worst team in the majors (43-67).
The Indians pounded out 16 hits in a 12-2 win Friday, but couldn't do anything with Texas starter Miles Mikolas, who went into the game with an 8.54 ERA, through five innings.
That finally changed in the sixth. Jason Kipnis started the rally with a single, took second on a sacrifice and went to third on a groundout. The Rangers elected to walk Carlos Santana intentionally on a 2-0 count, but Chisenhall foiled that strategy by dropping a single into center field on the first pitch.
Swisher lined a single to right for a two-run lead.
''Lonnie found some outfield grass and Swish got the hit,'' Francona said.
''He tried to get in on Chiz and he jammed him, but there was just enough there for him to hit,'' Rangers manager Ron Washington said. ''That was a tough break for Miles.''
T.J. House, called up from the minors before the game, pitched five scoreless innings for Cleveland.
House, who struck out seven and allowed three hits, was pulled in the sixth when Elvis Andrus drew a leadoff walk and took second on Alex Rios' infield hit. Atchison retired Adrian Beltre on a groundout that moved both runners into scoring position.
Andrus was tagged out in a rundown trying to score on Arencibia's grounder. Martin struck out on a 3-2 pitch to end the inning.
Mikolas (1-4) allowed two runs in seven innings.
Francona led his players and coaching staff to the pregame ceremony in which the Indians unveiled a statue of Jim Thome. As an opposing manager, Francona always respected Thome's power. ''If you go to a game, you don't really want to go get something to eat when he's hitting,'' Francona said. His strategy to stop Thome seldom worked. ''When he walked to the plate I'd gave him the bunt sign,'' Francona said. ''He'd always smile and hit one in the upper deck.''
TAKE IT INSIDE
The plan for Indians outfielder Michael Bourn (strained left hamstring) to run the bases was scrubbed by an afternoon thunderstorm. So he was sent across the street to Quicken Loans Arena, where the Cleveland Gladiators were hosting an Arena League playoff game. Bases were set up on the carpeted playing surface and Bourn was able to get in his work before the football game.
Rangers: Catcher Geovany Soto (strained right groin) was scheduled to catch seven innings Saturday on his rehabilitation assignment with Triple-A Round Rock. He went 1 for 2 with a walk Friday in his debut with the Express. Soto has only played in three major league games this season after undergoing right knee surgery in late March.
Indians: Outfielder Chris Dickerson missed his second straight game with a sore left knee, but Francona said he will probably be able to play Sunday. Dickerson left Thursday's game when he made an awkward attempt to catch a fly ball that fell for a double.
Rangers: Right-hander Yu Darvish (10-6) will start the finale of the three-game series on Sunday. Darvish's 1.24 ERA in day games is the second-lowest in the AL, while his 2.90 season ERA ranks ninth.
Indians: Right-hander Trevor Bauer (4-5) starts for Cleveland. He faced Texas for the first time in his career on June 6, allowing four earned runs in 6 1-3 innings and taking the loss.