Bumgarner (10-4) struck out eight and walked two en route to his first career shutout for San Francisco, which was coming off a three-game sweep of the Los Angeles Dodgers. The left-hander has made six starts in June and sports a 5-0 mark.
The Giants, who did not allow a run in their set with the Dodgers, have posted four consecutive shutouts for the first time in franchise history. The 36.0 consecutive scoreless innings streak is the longest in franchise history since July 26-30, 1948 (also 36).
"Amazing! It really is," Giants manager Bruce Bochy said of the shutout streak. "I've never seen it."
Ryan Theriot, Gregor Blanco, Pablo Sandoval and Angel Pagan had an RBI apiece for San Francisco, which improved to 4-0 on a seven-game homestand. The Giants are now in sole possession of first place in the NL West for the first time this season.
Ryan Hanigan had the lone hit for the Reds, who started an 11-game road trip on Thursday.
Johnny Cueto (9-4) allowed three runs -- two earned -- on six hits over six frames to take the loss. He fell to 4-1 in June.
The Giants gave Bumgarner all the support he would need in the first inning. Melky Cabrera worked a two-out walk, moved to second on Buster Posey's single and crossed the plate on Pagan's base hit to right. Jay Bruce's throwing error allowed Posey to score as well.
Cincinnati's first hit of the game came in the sixth, when Hanigan led off with a single up the middle.
"You feel helpless sitting over there. There's not much you can do when you're not getting hits and not getting baserunners," Reds manager Dusty Baker said.
San Francisco tacked on another run in the bottom of the sixth. Pagan worked a two-out walk and scored on Sandoval's double to left. Todd Frazier misplayed the ball off the bat and was caught out of position.
The Giants added two more runs in the seventh to build a 5-0 margin. Sam LeCure gave up a one-out single to Bumgarner, who raced home on Blanco's triple to right-center field. Theriot followed with an RBI single up the middle.
Bumgarner was perfect after giving up the hit to Hanigan.
"Madison was in charge the whole way. I actually thought he got a little better as the game went," Bochy said.
Bumgarner is the first Giants pitcher to win five games in a calendar month since Tim Lincecum did so in September 2010. He also improved to 1-2 in four career starts against the Reds...Cueto had lasted at least seven innings in his previous five straight starts...Reds second baseman Brandon Phillips, who turned 31 on Thursday, sat out with a head injury.
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