Clevenger, who was called up Friday from Triple-A Norfolk, got his first homer since June 29, 2012. He also had a career-high four hits in Baltimore's 18-2 victory Sunday.
"It feels great, especially this time of year," Clevenger said. "Right now, we're in the playoff race. Anything I can do to help the team. We're on a good streak right now. We have to keep it going in the next series."
Tillman (9-7) allowed two runs on three hits with three walks and three strikeouts over seven innings. He has not lost since May 31.
"We need to get on a run here and play good baseball like we have been," Tillman said. "We're swinging the bats well. I feel like right now as long as our starters are giving the team a chance to win and get deep into the game, set up the bullpen how it needs to be set up, I think we will be ready."
Oakland went winless on its seven-game East Coast road trip.
Oakland left-hander Sonny Gray (12-5), who was scratched from his previous start with back spasms, allowed four runs (one earned) on seven hits with two walks and five strikeouts over 5 2/3 innings.
"I could definitely tell that I hadn't been on the mound in a while," Gray said. "But I felt fine."
Danny Valencia provided the A's with a 1-0 lead when he homered to lead off the second. Valencia has a hit in nine of 10 games since being claimed off waivers Aug. 3.
The A's entered the game leading the majors with 96 errors. Another miscue helped the Orioles take the lead in the fourth when a grounder by Chris Davis rolled through second baseman Eric Sogard's legs. Jonathan Schoop followed with a single and then Clevenger connected for the three-run shot.
"When you have your ace on the mound, you want to play clean defensively," A's manager Bob Melvin said. "When you are not scoring many runs, you want to give yourself a chance because you know when he's on the mound, runs are going to be tough to come by."
In the fifth, Sam Fuld was called out for running out of the baseline. He and Melvin were then ejected by home plate umpire Brian Knight for arguing the call.
JOSEPH LEADS THE WAY
The Orioles improved to 38-5 when Joseph gets at least one RBI. He is batting .342 with five doubles, a homer and seven RBIs in August.
Athletics: LF Coco Crisp, who returned from the DL on Aug. 3 after missing two months with a neck injury, was held out of the lineup because of hip and ankle soreness. Crisp collided with shortstop Marcus Semien on Sunday.
Orioles: Wieters has missed five straight games with a strained right hamstring, but he ran the bases Monday and could be available Tuesday, manager Buck Showalter said.
Athletics: Left-hander Felix Doubront (1-1, 4.59 ERA) gets the start as Oakland returns home to face the Dodgers and their ace, Clayton Kershaw.
Orioles: Kevin Gausman (2-4, 4.48 ERA) takes the mound in the series opener against the Mets. Gausman allowed three runs over seven innings with a career-high eight strikeouts in his last outing against Seattle. That performance was overshadowed by the Mariners' Hisashi Iwakuma, who threw his first career no-hitter.