Oakland moved into sole possession of the top AL wild-card spot, one game ahead of Boston.
Indians reliever Nick Wittgren (2-5) retired the first two batters in the 10th before intentionally walking Matt Olson. Lowrie followed with a double to left field, scoring automatic runner Elvis Andrus.
"Lowrie is a good hitter, no doubt, and he put a good swing on it," Cleveland acting manager DeMarlo Hale said. "I thought we had a better matchup with him by walking Olson, but those are choices you've got to make sometimes."
Lou Trivino (5-4) pitched a scoreless ninth and Andrew Chafin earned his first save of the season.
Athletics relievers Burch Smith, A.J. Puk, Yusmeiro Petit, Sergio Romo, Trivino and Chafin held the Indians hitless for the final six innings.
"The bullpen was great, we made some big plays in the field, and we had timely hitting," Smith said. "It was an all-around good game for us."
Cleveland carried a 3-2 lead into the eighth, but Oakland tied it off James Karinchak with one out. Starling Marte singled, stole second and came home on Lowrie's grounder.
Seth Brown homered, singled and stole a base for the Athletics, who began their second 10-game road trip of the summer.
Marte was 1 for 5, dropping his average with Oakland to .400 in 10 games since being acquired from Miami. He also stole his 30th base of the season.
"At least one facet of everything that Starling brings to the table comes through in every game," Athletics manager Bob Melvin said. "We're thrilled to have him."
Amed Rosario and Myles Straw each had two hits for Cleveland - Rosario got the 500th of his career. The Indians are a season-high 10 1/2 games out of first place in the AL Central.
Triston McKenzie extended his home winless streak to eight starts, allowing two runs in six innings while striking out five. The right-hander's only victory at Progressive Field occurred in his MLB debut against Detroit on Aug. 22, 2020.
"I feel like we're putting ourselves in position to win ballgames," McKenzie said. "As starting pitchers, we want that, to give the team a chance to come out on top."
Athletics left-hander Sean Manaea made his shortest start of the season at 1 2/3 innings, giving up three runs and five hits on 69 pitches.
Josh Harrison doubled and scored for Oakland. He left in the sixth after being struck by a pitch on the top of his left hand, and X-rays were negative.
Athletics left fielder Mark Canha exited in the 10th with irritation in his right eye.
The Athletics played in Cleveland for the first time since May 22, 2019, a span of 811 days.
ROCK OF AGES
A statue of Indians great Rocky Colavito was unveiled earlier in the day in the Little Italy neighborhood of Cleveland. The fan favorite celebrated his 88th birthday at Progressive Field, receiving warm applause when introduced during the game.
Colavito played eight seasons in two stints for the Indians, hitting 190 of his 374 career home runs with the franchise. The nine-time All-Star led the AL with 42 homers in 1959, but was infamously traded to Detroit for Harvey Kuenn the next year.
Former Tigers teammates Denny McLain and Willie Horton accompanied Colavito to the ceremony.
MOVING ON UP
Puk -- the A's top pitching prospect -- was recalled from Triple-A Las Vegas and worked one scoreless inning. The 6-foot-7 lefty posted a 1.64 ERA in 13 appearances with the Aviators since June 26, showing no signs of the arm troubles that plagued him for the past year. "A.J. is here as a reliever," Melvin said. "We're good with our starters and (RHP Daulton) Jefferies would be the next in line."
Athletics: RHP James Kaprielian (right shoulder soreness) reported no physical problems after working six innings Sunday in a 6-3 win over Texas. Kaprielian had been on the 10-day injured list from July 28 to Aug. 7 with a shoulder impingement.
Indians: C Roberto Pérez (right shoulder inflammation), who was placed on the 10-day IL on Aug. 4, is eligible to be activated this weekend when Cleveland visits Detroit. Pérez missed 52 games earlier this season with a broken right ring finger.
Athletics: RHP Frankie Montas (9-8, 4.10 ERA) has 45 strikeouts in his last 31 2/3 innings, the most by an Oakland pitcher over a five-game span since Todd Stottlemyre had 46 in 1995.
Indians: RHP Cal Quantrill (3-2, 3.14 ERA) has only allowed three earned runs and 19 hits during his last five starts, posting a 2-0 record with a 0.90 ERA and 28 strikeouts in 30 innings.
Jed Lowrie delivers go-ahead run for A's in 10th
Jed Lowrie's double in the 10th puts the Athletics up 4-3 on the Indians.