Garrett Richards (11-8) tied a career high with 11 strikeouts in 7 2/3 innings. Huston Street got the last three outs for his 26th save in 29 attempts, and his first since becoming the 27th member of the 300-save club on July 22.
Gillaspie connected for a two-run drive in the sixth. The Angels added another run later in the inning for a 5-3 lead.
Kluber (6-12) gave up five runs and 10 hits in 5 2/3 innings, and left with a 3.60 ERA this season.
Kluber faced a lineup missing slugger Albert Pujols, who got the night off after playing first base in all three interleague games at Dodger Stadium because the Angels weren't able to use a designated hitter.
Gillaspie, acquired by the Angels from the Chicago White Sox in a cash deal on July 24 after third baseman David Freese broke a finger, connected after a leadoff single by Erick Aybar. It was his fourth home run this season and first in seven games with his new club.
Four batters later, David DeJesus -- one of three outfielders the Angels got at the trading deadline -- chased Kluber with a two-out RBI single.
The Indians scored three times in the first. Michael Brantley hit an RBI single and Gomes hit his sixth homer.
Cleveland didn't get another hit until Santana's two-out single in the sixth, which ended a string of 14 straight batters retired by Richards.
Indians: Placed All-Star 2B and leadoff hitter Jason Kipnis on the 15-day disabled list because of chronic inflammation in his right shoulder. He returned Cleveland to be examined by team doctors. ... RF Lonnie Chisenhall left the game in the fourth inning because of a cervical strain.
Angels: RHP Jered Weaver, who hasn't pitched for the club since June 20 because of inflammation in his left hip, is scheduled to make his second rehab start for Class A Inland Empire on Tuesday.
Indians: RHP Carlos Carrasco (11-8) is 7-2 with a 2.82 ERA in 10 road starts this season. Last Thursday at Oakland, he pitched a two-hitter and retired his final 16 batters in a 3-1 victory.
Angels: RHP Matt Shoemaker (5-7) has allowed fewer than three runs in each of his last five starts, and is coming off back-to-back scoreless outings against Minnesota (six innings) and Houston (seven innings).