Boston, on a tear since a season-opening 1-5 road trip, can complete a series sweep in Monday's traditional Patriots' Day game that starts at 11:05 a.m. Last year the Red Sox were just 2-7 at home against the Rays.
Ortiz, who signed a one-year contract in the offseason, is off to one of the best starts of his career.
He's 16 for 36 in nine games in April, a month in which he has struggled in recent years. He had 20 hits in 22 April games in 2009, eight in 16 games in 2010 and 23 in 25 games in 2011.
But on Sunday with the score tied 4-all in the sixth inning, he doubled home Kevin Youkilis, who had walked.
Ortiz went 4 for 5 with a homer and five RBIs in Saturday's 13-5 win. Then he doubled, singled and doubled in his first three at-bats on Sunday before grounding out to first to end the seventh.
Also in the seventh, Mike Aviles hit his second solo homer in two days from the leadoff spot. He moved there for the first time this season on Saturday when Jacoby Ellsbury went on the 15-day disabled list with a partially separated right shoulder.
Vicente Padilla (1-0) pitched 1 2-3 scoreless innings after Felix Doubront left the game after allowing Luke Scott's leadoff homer that tied it in the sixth.
Alfredo Aceves pitched the ninth for his second save in three opportunities.
Matt Moore (0-1) gave up six runs on eight hits in 6 1-3 innings in the third regular-season start of his career.
The Red Sox scored first for the first time in the series with three runs in the second. Youkilis singled, Ortiz walked and Cody Ross hit his second homer in two days. They made it 4-0 in the fourth on Kelly Shoppach's RBI double.
Doubront, making just his fifth major league start, allowed four hits in four scoreless innings before struggling in the fifth, when he allowed three runs.
Chris Gimenez singled, Desmond Jennings doubled and Carlos Pena doubled in two runs after having his shot into the right-field stands ruled foul by video review. Evan Longoria made it 4-3 with a ground-rule double in the fifth and Scott's second homer in two days tied it.
NOTES: Ralph Branca, father-in-law of Boston manager Bobby Valentine, threw out the ceremonial first pitch on the 65th anniversary of Jackie Robinson breaking baseball's color barrier. Branca, a right-handed pitcher, is the last surviving member of that Brooklyn Dodgers team in 1947. ... The Red Sox called up OF Jason Repko and INF-OF Nate Spears from Pawtucket, optioned OF Che-Hsuan Lin to the Triple-A team and designated for assignment RHP Michael Bowden and C Luis Exposito. ... Jennings snapped an 0-for-11 slump with a double in the third. ... The Red Sox have scored at least two runs against the Rays in seven different innings. Last year they scored two runs or less in eight full games against the Rays.