Ortiz hit a two-run homer and added a bases-clearing double while Ross contributed a two-run shot for the Red Sox, who won for only the third time in eight games to begin the season.
Mike Aviles, Dustin Pedroia and Jarrod Saltalamacchia also went deep as the Red Sox banged on 15 hits in the victory.
Clay Buchholz (1-0) earned the win despite allowing five runs on six hits in a seven-inning effort. He walked three and struck out five.
Rays starter Jeremy Hellickson was touched for five runs on seven hits over five innings, but it was Burke Badenhop (0-1) who was saddled with the loss after Tampa Bay had built an early 4-0 lead.
Luke Scott cracked a three-run homer and added an RBI double for Tampa Bay.
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