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.