With one out in the 10th, Jacoby Ellsbury stroked single against Brandon Gomes (0-1), stole second and then took third on catcher Jose Lobaton's wild throw.
Victorino did smack a ground ball, but shortstop Yunel Escobar needed to make a diving stop to reach it, allowing Ellsbury to score the game-winner.
David Ross smacked a fifth-inning solo homer to provide the other run for Boston, which took the opener of a rain-shortened three-game series.
Junichi Tazawa (2-0) earned the win with a scoreless top of the 10th.
Rays starter David Price yielded one run on four hits while striking out eight and walking two.
Sean Rodriguez provided Tampa Bay's only RBI.
Notable deaths of 2014 [PHOTOS]