Cave gave the Twins a 3-2 lead with his second homer of the season in the fourth inning. He added a two-run double in the fifth to cap a five-run rally that put Minnesota on top for good.
Cave tied Saturday night's game with a two-out single in the ninth before the Twins won in the 10th. The backup outfielder was recalled from the minors in early August and is seeing more playing time with All-Star center fielder Byron Buxton on the injured list. But he said he's ready to play any role manager Rocco Baldelli has in mind for him.
"If I get a chance to start, awesome, I love it," Cave said. "If I've got to come off the bench and help somebody out, then I love that, too."
The Twins outscored the Giants 20-5 in the series and have won three straight following a six-game losing streak that knocked them out of first place in the AL Central. The victory Sunday moved Minnesota within two games of first-place Cleveland, which got shut out in Seattle.
"I'd say this is probably one of our best series of the year. I don't know any other way to boil it down," Baldelli said. "The last three games is as good a three games as you can expect a team to play. Whether we were putting up runs or playing close ballgames, staying in the game, pitchers coming in and doing their jobs, complete baseball. That's what you want."
Michael Fulmer and Jorge López each pitched a perfect inning to end it.
Giants starter Jakob Junis (4-4) gave up six earned runs and nine hits in 4 1/3 innings. The outing completed a rough month for Junis, who went 0-2 with a 7.13 ERA in five August starts.
Brandon Crawford singled three times and drove in a run for the Giants, who have lost eight of 10 and ended a 2-6 road swing.
"Not good enough, not up to anybody in this clubhouse's standards," said manager Gabe Kapler, whose team is 7 1/2 games out of the final wild-card spot in the NL. "We have to come back home and keep our heads down, keep grinding, keep working every day and understand that there's still time. And really fight and scratch and claw to stay in this thing."
Carlos Correa went 2 for 4 with a double and an RBI for Minnesota. He was 8 for 12 with two runs and four RBI in the series. The star shortstop hasn't delivered the huge numbers some expected when he signed the largest free-agent contract in team history in March. But Baldelli sees signs that Correa is heating up at the right time.
"Carlos is very much locked in right now. I think he's taking control of the at-bats when he steps up to the plate," Baldelli said. "I think he knows what he's trying to do and he's simply doing it. He's doing it over and over again. When you get consistently good at-bats like that, you know it's all coming together in a really nice way."
Giants: 3B Evan Longoria left in the fifth with a tight right hamstring. He said he tweaked it while running out a double in the top of the inning. "I'm hoping that it's not anything serious," Longoria added. "I don't think it is."
Twins: 2B Jorge Polanco (knee) remained sidelined after he was removed from Saturday's game with patellar tendinitis.
Giants: LHP Carlos Rodón (12-6, 2.81 ERA) will face the Padres on Monday in the opener of a three-game series at home. The two-time All-Star is 4-0 with a 1.44 ERA in his last five starts. He threw a three-hitter with 12 strikeouts in a 3-1 victory at San Diego on July 9.
Twins: RHP Dylan Bundy (7-6, 4.56) takes the mound Monday as Minnesota opens a three-game series with the Red Sox at Target Field. In their first meeting this season, Bundy gave up one run in 5 1/3 innings on April 18 as the Twins won 8-3 in Boston.