Ortiz homered in the second and eighth innings as the Red Sox followed up their 12-2 win over the Rays on Friday with another victory at Fenway Park to lead the best-of-five divisional series 2-0.
Boston's Jacoby Ellsbury had three hits and scored three runs while Dustin Pedroia drove in three runs.
John Lackey took the win despite giving up four runs in 5-1/3 innings before the bullpen finished off the Rays.
Tampa's Cy Young winner David Price took the loss after giving up seven runs.
Games three will be held in Florida on Monday with the Rays needing to win to keep the series alive.