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Josh McCown made a bright start to the game, connecting with several receivers to move the hosts into the red zone before running the ball in himself.
During the second quarter, the game had to be halted for over an hour as thunderstorms moved into the Tampa area, forcing the players to head for cover.
Once play resumed, the Rams quickly got into field-goal range, allowing Greg Zuerlein to split the posts from 36 yards out and give St Louis the lead at half time.
After using Vincent Jackson to set up another scoring opportunity, McCown was on the move again in the third, darting in from one yard out for his second rushing touchdown of the day.
Two more field goals from Zuerlein restored the the visitors' advantage, while Tampa Bay rookie kicker Patrick Murray saw his first field-goal attempt as an NFL player blocked.
Leading by just two points with moments to spare, McCown tossed the ball to Mike Evans to set the hosts into field goal range, but the wide receiver was injured forcing a mandatory 10-second run off, leaving the Bucs without enough time to bring on the kicker.