A day after Tim Lincecum outdueled Roy Halladay in a marquee matchup of aces, Oswalt beat Jonathan Sanchez.
The series shifts to San Francisco for Game 3 on Tuesday afternoon. Matt Cain faces Philadelphia's Cole Hamels, the 2008 World Series MVP.
Oswalt allowed one run and three hits, striking out nine. He also had a hit and scored a run on a mad dash through the third base coach's stop sign.
Cody Ross hit his third solo homer in two games for the Giants.
Rollins busted out of a 1 for 15 postseason slump, going 2 for 3 with a bases-loaded walk and a bases-clearing double.