Bumgarner, who pitched a one-hitter against the Reds on June 28, retired the first three hitters before Ludwick's home run to center quieted the sold-out crowd at AT&T Park. It is only the second home run given up by Bumgarner in five postseason appearances.
Ludwick was also instrumental in the Reds three-run fourth.
Joey Votto and Ludwick both singled to open the inning. After Bumgarner got Jay Bruce to foul out to third baseman Sandoval, Scott Rolen singled to drive in Votto and took second when San Francisco right fielder Pence threw to the plate.
Ryan Hanigan followed with a two-run single to center to make it 4-0.
"Being able to get Cain and Bumgarner out that early and get to the bullpen was important. Everybody knows how good they are," Reds outfielder Drew Stubbs said. "We came out here with the intention of getting one of two. Getting two is a bonus. Now the ball is in our court so to speak and we have to finish it out. This is who we are and what we are capable of."
Bochy pulled Bumgarner after he allowed consecutive hits in the fifth. George Kontos replaced Bumgarner and got Ludwick to ground into a 6-4-3 double play.
Lincecum, the long-haired star of the pitching staff during the Giants improbable title run, made only the third relief appearance of his career and allowed one hit over two scoreless innings.
And the rest of the relievers who had been so reliable after losing All-Star closer and 2010 majors saves leader Brian Wilson to Tommy John surgery couldn't shut down the Reds, either.
Fans left the Giants waterfront ballpark in droves in the midst of Cincinnati's five-run eighth inning, unsure whether their team would be back for another game this season.
Homer Bailey, who pitched a no-hitter Sept. 28 at Pittsburgh, will pitch for the Reds on Tuesday.
"We're not done yet," Bumgarner said. "It's happened before to teams and they've come back and won. It just doesn't happen very often but it has been done before so we're going to keep fighting until the end."