The Red Sox gave up 10 runs in the sixth inning and lost 14-6 to the Miami Marlins, who completed a two-game sweep in a series between last-place teams.
The afternoon game drew a crowd of 26,041, with more than half rooting for the Red Sox. Ortiz drew the biggest cheer of the day with his first homer, which landed 10 rows into the upper deck.
"It felt like I was playing in the Dominican," he said. "It feels good to feel like you're playing at home."
Ortiz has 25 homers this season and 491 in his career. He has started fielding questions about his chances of reaching the 500 milestone this year.
"I don't know," he said. "I'm just going to keep on swinging. I'm not trying to put any pressure on myself and just trying to put a good swing on the ball and not wasting any swings at the plate. So hopefully."
His 48th multi-homer game was his second this year.
"He's a great hitter," manager John Farrell said. "Anybody approaching the 500 mark has done a lot of good things for a lot of years, and David is certainly in the midst of a very strong year once again."
Rodriguez (6-5) allowed eight runs in five-plus innings. After pitching scoreless innings in the third, fourth and fifth, he gave up four consecutive hits to start the sixth.
"Surprising the sixth inning played out like it did with Eddie on the mound, given the three previous innings," Farrell said.
Rookie Kyle Barraclough (1-0) pitched 1 1-3 scoreless innings for his first major-league victory.
Rookie J.T. Realmuto drove in six runs, including five in the sixth, when Miami's 10 runs tied a franchise record.
Realmuto hit an RBI single in the sixth to put the Marlins ahead 5-4, and then hit his first career grand slam later in the inning to make the score 14-4.
He's the first Marlins player to drive in five runs in an inning. It was the second time the Marlins reached double figures in runs in an inning — they scored 10 times at Arizona in 2009.
Realmuto also drove in a run with his sixth triple, the most by a rookie catcher since Carlton Fisk of the Red Sox in 1972, according to STATS. Needing a double in the eighth to hit for the cycle, he struck out.
The only catcher with more than six RBIs in a game against the Red Sox was Yankees Hall of Famer Yogi Berra in 1957, according to STATS.
"Anytime you can be spoken in the same sentence as those greats, it's huge," Realmuto said. "Obviously we shouldn't use my name in there too often just yet."
Ichiro Suzuki drove in three runs with a single and a triple for Miami. Suzuki nearly tripped over Ortiz at first base running out a two-run triple in the sixth.
The teams came into the series with the worst records in their leagues. The Marlins won their third in a row.
Red Sox: OF Hanley Ramirez will be examined by a doctor Thursday when the team returns to Boston to determine whether further tests are needed on his sore left foot. ... RHP Clay Buchholz (right elbow strain) received a favorable evaluation Wednesday from Dr. James Andrews and will begin a throwing program. ... RHP Rick Porcello (sore right triceps) is scheduled to make two rehab starts, beginning Saturday for Class-A Lowell.
The Red Sox begin a 10-game homestand Friday when Joe Kelly (4-6, 5.96) starts against Seattle's Mike Montgomery (4-4, 3.25).