Allen (7-7) allowed one run on three hits and struck out six to get his first victory since Aug. 15. The left-hander is fourth among AL rookies in strikeouts (113) and fifth in wins.
Emmanuel Clase retired the side in the ninth for his majors-leading 39th save.
The Guardians remained 6 1/2 games behind Minnesota in the AL Central after the Twins beat the Mets on Friday.
“There’s no doubt that our backs are against the wall and the odds are against us, but all we can do is worry about winning that day,” Allen said. “We were good tonight and have to keep it up.”
Logan O’Hoppe had a two-run homer in the sixth inning for the Angels, who have dropped seven of their last eight.
Griffin Canning (7-6) allowed three runs on eight hits in 5 1/3 innings.
“I thought he threw the ball very well. He was at 92 pitches in the sixth and also kind of looking for a nice left-handed lane for (Kolton) Ingram (out of the bullpen) and it didn’t work out,” Angels manager Phil Nevin said of Canning.
Josh Naylor put the Guardians on the board in the third with a base hit up the middle to drive in Steven Kwan, who led off the inning with a double.
Naylor’s other two RBIs came when he was walked with the bases loaded in the sixth inning and hit a sacrifice fly in the eighth.
Bo Naylor had two hits, including an RBI single in the fourth inning to extend Cleveland’s lead to 2-0. The catcher is 15 for 41 with 10 RBIs in his last 15 games.
Cleveland’s other runs came on a sacrifice flies by Kwan in the sixth and Ramon Laureano in the eighth.
O’Hoppe hit a two-run homer to center on a hanging sinker by Nick Sandlin in the sixth to get the Angels within 3-2. They had the tying run on third with two outs, but Sam Hentges struck out Kyren Paris to end the rally.
Angels DH Shohei Ohtani was out of the lineup for a fifth straight game due to right oblique tightness.
Manager Phil Nevin said he thought it was best if Ohtani had another day off with the Guardians having a left-hander on the mound in Allen.
“Obviously, he’s our best either way. If he feels good tomorrow, tomorrow would be the best day to bring him back,” Nevin said.
Guardians: Manager Terry Francona said RHP Shane Bieber (right elbow inflammation) did well facing live hitters on Thursday in Cleveland. The next step in Bieber returning to the rotation would be a rehab assignment.
Angels: 2B Luis Rengifo will undergo further tests after he strained his left biceps while taking a swing in the on-deck circle Thursday night.
Guardians: RHP Lucas Giolito (7-12, 4.88 ERA) allowed nine runs in three innings in his first Cleveland start on Monday. He was claimed on waivers from the Angels on Aug. 31 after he went 1-5 with a 6.89 ERA in six starts after being acquired from the White Sox.
Angels: LHP Tyler Anderson (5-6, 5.55 ERA) had a no-hitter through five innings last Sunday at Oakland before allowing three runs on three hits in a 10-6 loss.