The Red Sox had seemed on their way to a win in the 10th when Jarrod Saltalamacchia delivered a two-run single, but the Rays came back with two runs of their own in the next half-inning off Andrew Bailey, extending the contest.
Then in the top of the 14th, Shane Victorino began the frame with a single off Cesar Ramos (1-2) and went to second on an out.
The Rays intentionally walked David Ortiz to get to Nava, but he crossed up their strategy by blooping a hit into center field, plating Victorino with the go-ahead run.
Saltalamacchia (4-for-7) added an RBI single for an insurance tally and the American League East-leading Red Sox took the opener of a three-game series.
Franklin Morales (2-0) tossed two scoreless innings to earn the decision.
Ben Zobrist had five hits for Tampa Bay, which rallied from a six-run deficit to force extra innings.