The Red Sox claimed the best-of-five Division Series 3-1 and they will host either Oakland or Detroit in a best-of-seven set scheduled to start on Saturday in Boston for the right to represent the American league in the World Series.
The A's and Detroit Tigers are tied 2-2 in their series with the deciding Game Five to be played in Oakland on Thursday.
Boston squandered numerous scoring chances early, including a bases loaded, no-out opportunity in the second inning, and overcame a 1-0 deficit late in the game against a merry-go-round of Rays pitchers.
The Red Sox scored two runs in the seventh on a wild pitch by reliever Joel Peralta and an infield single by Shane Victorino before adding an insurance marker in the ninth after closer Fernando Rodney walked two, threw a wild pitch and hit a batter to set the stage for a sacrifice fly by Dustin Pedroia.
Tampa Bay desperate to avoid elimination, used nine pitchers, every one they had on their roster.
Boston closer Koji Uehara bounced back from a rare failure on Monday when he served up a game-ending home run to Jose Lobaton, by registering a four-out save, striking out Rays slugger Evan Longoria to end it.
Craig Breslow, the second of three Boston relievers used to follow starter Jake Peavy to the mound, struck out all four batters he faced and registered the win.