The Giants did not scrape out their first run of the game until the seventh and it came with the help of a bunt by Gregor Blanco that hugged the third base line and rolled to a stop in fair territory.
San Francisco won its fifth straight game in the playoffs and will try to move closer to its second World Series title in three years when the action resumes in Detroit on Saturday. The Tigers will host the next three games, if that many are needed.
Bumgarner turned into an unexpected hero after giving up 10 runs on 15 hits in just eight innings during his two previous post-season starts this year. In his last nine starts, including the two in the playoffs, his earned run average had been 6.85.
Detroit starter Doug Fister was almost as effective as Bumgarner, blanking the Giants through the first six innings. He finally allowed a run in the seventh, however, while San Francisco was hitting only one ball out of the infield.
Hunter Pence opened the seventh with a single to left and Brandon Belt fouled off two pitches before drawing a walk on a 3-2 count.
That brought up Blanco, the Giants' left fielder who is a career .253 hitter and who strikes out an average of once in every four at-bats.
Blanco was under orders to bunt, which he tried to do on a 2-0 pitch only to foul it off. He tried again on Fister's next offering and it was perfect, trickling down the third base line.
Detroit catcher Gerald Laird let the ball go in hopes that it would roll foul, but it never did.
With the bases now loaded, Detroit Manager Jim Leyland chose to play the infield back -- conceding a run in exchange for a possible double play. That is what happened when Brandon Crawford hit a grounder to second base.
The Tigers got their double play, but San Francisco got their run when Pence raced home from third.
Pence then provided the Giants with an insurance tally in the ninth with a sacrifice fly, although San Francisco left the bases loaded and missed a chance to open a big lead.
Ryan Vogelsong, with a 2-0 record during this year's playoffs, is scheduled to start for the Giants in Game 3 on Saturday. Detroit will counter with Anibal Sanchez, who allowed three hits in winning the second game of the American League Championship Series against the New York Yankees.