The win, coupled with the Dodgers' loss to the Mets, moved the Giants into sole possession of first place in the National League West for the first time since last Aug. 9.
In his first career complete game, Bumgarner didn't give up a hit until the sixth inning, and Giants starting pitchers have now gone a San Francisco-era record 36 straight innings without allowing a run.
"It's nice to finally get a complete game," Bumgarner said afterward, still dripping from a postgame celebration in which he was doused with shaving cream and beer by his teammates. "I had a little luck on my side, but I'll take it."
After Barry Zito, Ryan Vogelsong and Tim Lincecum each pitched seven scoreless innings in a three-game sweep of the Dodgers, Bumgarner followed with a masterful eight-strikeout performance to make the Giants the first team in the Major Leagues since the 1995 Orioles to register four straight shutouts.
"It's been quite an impressive run these guys have been on," manager Bruce Bochy said. "That's how good they are. They all have been locked in. Four consecutive shutouts -- I've never seen it, it's so hard to do."
"You have to have four pitchers on top of their game, and Madison was tonight," Bochy said. "That's quite the game he threw. He had all his stuff going, he hit spots. He pitched efficiently, he was outstanding. He had to be, because we were facing a tough guy."
The Giants' offense backed Bumgarner by giving him an early 2-0 lead. Angel Pagan's single in the bottom of the first plated Melky Cabrera, and Buster Posey also came around to score when Jay Bruce's errant throw to home from right field got past catcher Ryan Hanigan.
"He was aggressive there," Bochy said of Posey. "He made the turn at third base, saw what happened and kept going."
Another defensive mishap aided the Giants in the sixth when Reds left fielder Todd Frazier misplayed Pablo Sandoval's line-drive RBI double, initially taking steps forward before watching the ball fly over his head.
Reds starter Johnny Cueto entered the game with a career 1.64 ERA against the Giants, but was tagged for two runs on six hits in six innings. All nine Giants batters reached base Thursday night, eight of them on hits.
"I think they are getting much better at-bats," Bochy said. "They're really riding out their at-bats and doing some good things. They've seen more pitches, I think that's been one of the keys to our success."
Bumgarner even got in on the offensive action, singling in the seventh and later coming around to score on Gregor Blanco's standup triple to left field.
Hanigan's single was the only hit the Reds got off Bumgarner on Thursday, and the 22-year-old left-hander's performance had his opponents tipping their caps.
"It was probably the best game he's ever pitched," Reds manager Dusty Baker said. "There's not much you can do about that."
The Giants' scoreless stretch matches one set by the 1948 New York Giants squad, which threw 36 straight scoreless innings from July 26-30. On Friday, the ball will next be handed next to Matt Cain, who got a friendly reminder of the rotation's streak from Bumgarner.
"I said don't worry about it, all you have to do is throw a shutout," Bumgarner said.