The A's could potentially find themselves in first place with a sixth straight home victory over the Indians on Saturday night.
Oakland (69-52) was outscored 21-8 while being swept in a four-game set at Cleveland in May, but it scored just enough to beat the Indians 3-2 at home Friday.
The A's second straight victory overall and fifth in a row at home over Cleveland (65-57) leaves them a half-game behind Texas in the West. Oakland last sat alone atop the division Aug. 6.
"Every one's huge now,'' A's infielder Eric Sogard said. "We're not in the middle of the game looking at the scoreboard, though.''
Yoenis Cespedes hit a two-run homer in the first inning and Sogard delivered a go-ahead double in the seventh after ex-Athletic Nick Swisher tied the score in the top of the frame with an RBI single.
Cespedes is 10 for 25 with two home runs and eight RBIs in seven career home games against the Indians.
The A's hope Dan Straily (6-6, 4.19 ERA) can build on the longest outing of his short career when he takes the mound Saturday. The right-hander is 0-4 with a 3.95 ERA in five starts since the All-Star break after giving up a run in 7 1-3 innings before Oakland's offense erupted in a 5-1 win Monday.
He had not completed five innings in each of his previous three outings.
"The mindset was throwing as hard as I can for as long as I can," Straily told the Athletics' official website. "I was able to execute pitches, keep the ball down in the zone, and that's something I haven't been able to do the last few times out."
This will be his first appearance against the Indians, who counter with Ubaldo Jimenez (8-7, 4.11).
Jimenez gave up five runs - two earned - and nine hits over four innings of a 10-0 loss at Miami on Aug. 2, then bounced back to last 6 1-3 while allowing three runs in a 7-2 loss to the Los Angeles Angels last Saturday.
The right-hander improved to 2-0 with a 3.57 ERA in three starts against the A's when he allowed two runs and struck out eight in 5 2-3 innings of a 7-3 victory May 6.
Swisher, who still gets booed in Oakland, had two hits for a Cleveland team that trails first-place Detroit by six games in the Central and is 4 1/2 back of the A's for the final wild-card spot.
"We couldn't catch a break,'' losing pitcher Justin Masterson said of the Indians, who stranded 13 and went 1 for 9 with runners in scoring position. "We were getting some good at-bats. We got guys on, we weren't able to get that extra one.''
Swisher is 6 for 18 versus Oakland this season, and batting .343 in his last 17 at the Coliseum.
Teammate Asdrubal Cabrera is hitless in his last 20 at-bats.