SAN FRANCISCO -- For nine innings Tuesday night, Conor Gillaspie got to watch a baseball game for his 30th birthday.

Then as a present, he was allowed to participate in the 10th inning, and that was when the party really started.

Gillaspie doubled as a pinch hitter off Cleveland Indians closer Cody Allen to open the 10th and scored two batters later on Eduardo Nunez's single, delivering the San Francisco Giants a 2-1 victory over the American League Central leaders.

"No wonder the ball got through," Giants manager Bruce Bochy said of Gillaspie's birthday, of which he was not aware. "The baseball gods took care of him."

The Giants earned only their fourth win in 12 interleague games this season while evening the three-game series at one apiece.

The Indians, the losingest team in the majors in interleague play, fell to 4-12 against National League competition this season.

It was their first extra-inning affair since last season, snapping a franchise-record, 84-game streak.

"Sometimes teams go through periods like this," Indians manager Terry Francona said of his club's fourth loss in five games since the All-Star break. "We've just got to figure out a way to win 2-1 as opposed to losing 2-1."

Gillaspie led off the 10th with a liner down the right field line that barely eluded Indians first baseman Edwin Encarnacion.

"I kinda stopped running," said Gillaspie, who thought the ball had been caught.

After Denard Span, attempting to sacrifice, bunted for a single, Nunez lined his single through a drawn-in infield and into right field off Allen (0-5). Pinch runner Kelby Tomlinson came home from third base with the game-winner.

The walk-off hit was the first of Nunez's career and the fourth by a Giant this season.

"I feel very grateful," Gillaspie said of his contribution to the win. "The stretch we've been in ... we've been kinda scuffling. To get a win like this, that's exciting."

Nunez had two hits for the Giants, who had lost four straight at home and three in a row overall.

Sam Dyson (1-1) worked two hitless innings to pick up the victory, which allowed San Francisco to avoid becoming the majors' second 60-game loser (Philadelphia).

Bradley Zimmer had two hits for the Indians.

Both teams had seven hits.

Starting pitchers Mike Clevinger and Ty Blach left a 1-1 game in the seventh inning. Neither got a decision.

Clevinger, facing the Giants for the first time in his career, was a bit unlucky not to leave with a lead. The only run he allowed was a direct result of a dropped fly ball by Indians right fielder Brandon Guyer, which resulted in a two-base error with one out in the sixth inning.

Clevinger struck out Brandon Belt for a second out before Buster Posey grounded a single up the middle, scoring Nunez from second base with the game-tying run.

"(Clevinger) was pitching really good and didn't deserve to give up any runs, so it stinks, man. It stinks," Guyer said. "I wish I could've made that play for him."

Clevinger allowed three hits and the unearned run in six innings. He walked two and struck out seven.

Blach, meanwhile, also might have avoided the only run he allowed. But in his case, it was his own fault.

Blach had a chance to force out Zimmer at second base and perhaps initiate a double play on Clevinger's sacrifice bunt attempt in the third inning, but the lefty couldn't immediately get a grip on the ball. He then had no choice but to throw to first, retiring Clevinger but allowing Zimmer to reach second.

Guyer followed with a triple to left-center, plating the game's first run.

Blach went seven innings in his first career outing against the Indians, allowing one run on seven hits. He walked one and struck out three.

NOTES: Neither team has won an interleague series this season. That streak will end for one of them Wednesday. ... Giants 3B Conor Gillaspie had been 0-for-5 on his birthday in his major league career before his 10th-inning double. ... The Giants improved to 9-4 in extra innings. ... The Indians' previous record for consecutive games without extra innings was 81, set during the 1901-02 seasons. ... Indians manager Terry Francona announced before the game that RHP Corey Kluber might have to miss his scheduled start Friday night against Toronto because of neck soreness. The All-Star is slated to throw a bullpen session Wednesday to assess his situation. ... If Kluber must be skipped or pushed back, Francona said the Indians would reinstate RHP Danny Salazar (sore right shoulder) from the disabled list and start him against the Blue Jays. Salazar stamped himself ready to return to the active roster by throwing six shutout innings for Triple-A Columbus on Monday night.