Gausman (7-4) allowed just one run and two hits as the AL East-leading Orioles got back in the win column after scoring just one run in their previous two games.
"It's been a rough couple of days, but those things are going to happen," said Steve Pearce, who had a leadoff double in the sixth and then homered in the seventh. "It was only a matter of time. We put together some good at-bats and grinded it out."
Hardy delivered a two-out RBI single and the slumping Davis added an RBI double to rally the Orioles, who were blanked over the first five by Danny Salazar (4-7).
Jonathan Schoop also homered for Baltimore, which is an MLB-best 28-13 since June 30. However, the Orioles were listless for 25 innings, scoring once Friday, not at all Saturday and they trailed 1-0 before taking their first lead of the series in the sixth -- thanks to Hardy and Davis.
Salazar, recalled before the game from Triple-A Columbus, held Baltimore without a run on four hits through five before running into trouble in the sixth. He gave up Pearce's double and hit Adam Jones before he was replaced by Scott Atchison, who got two quick outs.
Hardy, though, came through with his single and Davis, who was dropped again in the order by manager Buck Showalter, looped a soft double into left to make it 2-1.
Gausman then set down the Indians in order in the sixth before turning it over to Baltimore's bullpen.
The right-hander, who improved to 3-1 on the road, sensed the Orioles' urgency to leave town with a victory.
"I definitely knew we needed a win," Gausman said. "We hadn't played very good baseball the last two days. When you get walked-off [Friday's 2-1 loss] and then you get beat the next day, it's huge when you get that last game."
The Indians failed to take advantage of another strong start.
Pearce homered off C.C. Lee in the seventh and Schoop's leadoff drive in the ninth made it 4-1.
Although he took the loss, Salazar continued a torrid run of starting pitching for the Indians, who got a typically strong outing from Corey Kluber in the opener before Carlos Carrasco blanked the Orioles on three hits over seven innings.
Over the past seven games, Cleveland's starters have posted a 1.02 ERA -- five earned runs in 44 innings.
As much as he wanted the sweep, Indians manager Terry Francona was satisfied with taking two of three from the free-swinging Orioles.
"You hope they get out of town and beat up on somebody else because they were pretty quiet for two games and a half of another," he said. "But there's a lot of thunder in that lineup."
Davis began the series batting second, dropped to fifth Saturday and hit seventh in the finale. He snapped an 0-for-11 skid with a double in the third. It was his first game with two doubles since May 10, 2013.
The Orioles will spend the next week in Chicago. After playing three against the White Sox, they'll have an off day before opening a weekend series at Wrigley Field against the Cubs.
Orioles: 3B Manny Machado (sprained right knee) continues his rehab program, but he's not doing any baseball activities. There's no timetable for his return.
Orioles: RHP Bud Norris (10-7) will start the series opener against the White Sox and Chris Sale (10-2) as the Orioles continue their nine-game road trip.
Indians: After an off day, the Indians will open a three-game set in Minnesota with Trevor Bauer (4-7) facing Twins RHP Kyle Gibson (11-9).
Copyright by STATS LLC and The Associated Press
CLEVELAND, OH - AUGUST 17: Starting pitcher Danny Salazar #31 of the Cleveland Indians pitches during the first...