Prince Fielder also connected for a three-run shot as the Rangers (23-23) got back to .500 for the first time since the sixth game of the season. Fielder is batting .583 (14 for 24) with five homers and 15 RBI in his last five games.
Moreland's fifth homer came on a 2-0 pitch from Nick Hagadone (0-1) with one out.
Josh Hamilton, playing his second game of the season for Texas, was hitless in four at-bats and is 0 for 7 in his return to the Rangers.
Tolleson retired his first two batters, but shortstop Elvis Andrus booted Jason Kipnis' ground ball and Carlos Santana was hit on the wrist by a pitch. Michael Brantley then hit a tapper to the mound for the finale out.
The start of the game was delayed for 56 minutes because of rain.
Jason Kipnis and Ryan Raburn homered in the first for the Indians, who have dropped two straight after winning six in a row.
Indians starter Danny Salazar was in command before he walked Leonys Martin and hit Shin-Soo Choo with one out in the fifth. Fielder drove Salazar's next pitch into the seats in right for his 10th homer. He also leads the AL with a .371 batting average.
Salazar remained in the game after being treated for a finger problem on his right hand. He allowed six hits and walked three in 5 2/3 innings.
Texas starter Wandy Rodriguez gave up three runs in 5 1/3 innings.
Hamilton, acquired from the Angels on April 27, struck out in the first, grounded out in the fourth, flied out in the fifth and lined out in the eighth. The 2010 AL MVP is coming back from offseason shoulder surgery.
Rangers: No timetable has been established for the return of RHP Neftali Feliz, who had surgery for an abscess on his right side Saturday.
Indians: C Yan Gomes made his second start behind the plate since returning from a sprained right knee. Gomes, who was injured April 11, had a sacrifice fly and a single.
Rangers: RHP Colby Lewis will start the finale of a nine-game trip. He has been supported by seven or more runs in each of last four starts.
Indians: RHP Carlos Carrasco lost to the Rangers on May 17, allowing five runs in eight innings, including a two-run homer to Moreland.