Posey interrupted a 3-for-40 tailspin by leading off the sixth with his fifth homer of the year, a drive off Cleveland reliever C.C. Lee that landed halfway up the left-field stands.
The Giants have won the first two games of this series after posting the Majors' third-worst Interleague record (13-22) in the previous two seasons.
Reversing their fortunes has required a variety of contributions from the bullpen to the batting order to the bench.
Gutierrez's strikeout began a stretch of 4 1/3 hitless innings from Giants relievers, including Sergio Romo, who recorded his sixth save in as many chances. San Francisco's bullpen owns a Major League-low 0.45 ERA in 11 home games this season.
The Giants entered Saturday with a .138 batting average (13-for-94) with runners in scoring position during their previous 12 games. They collected two such hits in their big fifth, receiving pinch-hitter Gregor Blanco's RBI single before Pence came through.
"We have a good vibe," Pence said. "We have a team that's going to compete hard. ... Those are the kinds of wins and kinds of games that we feel we're going to bring to the ballpark every day. That's our effort."
The effort accelerated after Lincecum, who complained of inconsistent mechanics, issued two-out walks to Michael Brantley and Asdrubal Cabrera to fill the bases in the fifth. In came Gutierrez, who fanned Gomes with a 95 mph fastball. Though Gutierrez has a reputation for throwing hard, he said, "I know the guy [Gomes] was looking for a breaking ball with two strikes."
Freezing Cleveland's lead was essential, particularly since Indians starter Zach McAllister (3-1) had blanked the Giants on one hit through four innings.
"A base hit kind of puts the game out of reach," Giants manager Bruce Bochy said.
Michael Morse's leadoff single began the Giants' surge in the fifth. Brandon Crawford singled and Brandon Hicks walked while forcing McAllister to throw 15 pitches. That loaded the bases with one out for Blanco, another slumping Giant who connected on McAllister's first pitch to score Morse.
"The plan was be aggressive," said Blanco, who received that advice from coaches concerned with his .100 average.
After Angel Pagan's sacrifice fly delivered Crawford, Blanco sustained his forward approach by stealing second base. That put him in position to score along with Hicks on Pence's grounder, which scooted past second baseman Jason Kipnis.
"At that point, you have to be ready for his fastest pitch," said Pence, who hit a 95 mph fastball from McAllister.
Kipnis attempted to field the ball upright and appeared to have a chance at snaring it.
"I was planning on diving the whole time until I kind of gained ground on it," he said. "I caught up to it and thought I could reach it. It kind of caught me in between steps and it's one of those ones where you just don't know if you can bend down, slide or dive. I went for the reach and it scooted under my glove. ... It cost us pretty much the game."
The Giants' sixth began with Posey's homer, which ended his 0-for-24 skid against right-handed pitching. It also snapped a 27-inning homerless streak recorded by Cleveland's bullpen.
Posey thrilled the AT&T Park crowd by clobbering Lee's 3-2 slider.
"You're looking to battle anything close," Posey said. "Fortunately I had a good pitch I could handle."
Posey mentioned the importance of timing at the plate after he homered last Sunday at San Diego. Saturday, he explained the difficulty of maintaining that quality.
"You play this game every day. Every day's different," he said. "Your body feels differently. You're facing different pitchers. You play in different ballparks. You're playing day and night. Each day's not going to be the same. The big thing is making adjustments day to day."