Price (6-5) notched his fifth victory in six starts, taming the Bosox on just two hits in a game that was rescheduled following a Thursday rain-out.
The left-hander didn't issue a walk and struck out eight, tossing 67-of-90 pitches for strikes in the stellar effort.
Price exited in the eighth with a 2-1 lead and was replaced by Joel Peralta, who yielded a double to Ryan Lavarnway. Daniel Nava came in as a pinch-runner and went to third when Stephen Drew followed with another two-bagger.
That brought up Brandon Snyder, who flew out to left field. Nava came home after the catch and appeared to get his foot on the plate before being blocked by catcher Jose Molina, but home plate umpire Jerry Meals called him out.
Red Sox Manager John Farrell was ejected in a argument following the play.
Felix Doubront (7-5) absorbed the loss, yielding two runs on eight hits over five full frames.
Snyder socked a solo homer for the Bosox.