Cueto contributed his fair share as well, holding the Angels to no runs on two hits after carrying a no-hitter into the sixth inning. He lowered his ERA to 0.35 after four starts.
"It's great to get a series," Giants manager Bruce Bochy said. "You hate to wait this long to get one, but on the road against a club that has really been playing well this year, Johnny came through for us. What a great effort he gave us and we hung on."
The Giants won two of the three games at Anaheim, in a rematch of the thrilling 2002 World Series won by the Angels.
Cueto, who missed a recent start with an ankle sprain, pitched through some discomfort, even getting a visit from the trainer at one point. He did not give up a base hit until Ian Kinsler singled to lead off the sixth inning.
He closed the sixth by loading the bases but getting Luis Valbuena to hit into a 3-6-1 double play to end the threat. Cueto did a spin and made a fist pump after recording the out at first base, which ended up being the final out of his outing.
"Based on the situation I had to celebrate it," Cueto said.
A day after striking out 17 times against the Angels, the Giants tagged 77 pitches on Angels starter Jaime Barria in two-plus innings, with 49 of those pitches coming in the first inning alone. Barria gave up two runs on five hits with a walk in his second career appearance.
Belt's 21-pitch at-bat, as the second batter of the game, took about 13 minutes and included 16 foul balls. It ended with him hitting a fly ball to right field. He saw 17 combined pitches in his next two at-bats, and he finished those with a single and his home run, respectively
Belt ended the day with three hits and saw 40 pitches in his five plate appearances.
"I don't really remember [specifics], I just knew I was getting tired and I wanted to do something right there," Belt said. "I didn't want to give in. I put too much effort into that at-bat already. I wanted to get something to handle and he just kept making good pitches."
Buster Posey, who returned to the lineup after missing Saturday's game with back tightness, brought home the first Giants run in the third inning when he grounded into a double play. Evan Longoria added a two-run home run in the third, his fourth of the season.
Belt's towering home run in the fifth easily cleared the wall in right field and gave him four home runs in the past four games he has played. He did not play in Friday's game at Anaheim, which was also his 30th birthday.
"It was a hard-fought game, but no question it's good to head home with a series under our belt," Bochy said.
MOMENTS THAT MATTERED Cueto spun around like a top in celebration after working his way out of a bases-loaded situation in the sixth inning. After loading the bases with one out, Cueto got Valbuena to hit a ground ball to Belt at first base. Belt started a nifty 3-6-1 double play, with Cueto combining a spin with a pump of his fist following the last out of his outing.
SOUND SMART Cueto's 0.35 ERA is the lowest for a Giants pitcher after four starts since Ray Sadecki had a 0.25 mark in 1968. Cueto has given up one run over 26 innings so far this season, firing scoreless outings at the Dodgers, Diamondbacks and now the Angels.
HE SAID IT "You have to give it to him. He kept on throwing strikes and didn't really hang anything except maybe a couple of pitches, but it was enough to throw me off and not get the barrel to the ball. It was a tough at-bat, but I was just trying to have a good at-bat up there, honestly." -- Belt, on facing Barria in their 21-pitch duel in the first inning
UP NEXT The Giants will try to carry some momentum into the upcoming homestand, facing the Nationals on Monday. Chris Stratton (1-1, 2.22 ERA) will take the mound with some momentum of his own as the Giants have gone 3-1 in his starts this season. The Nationals will counter with left-hander Gio Gonzalez (2-1, 2.49) in the 7:15 p.m. PT start.