Johnny Damon had three hits, including a solo homer, and three RBIs, and John Jaso and Reid Brignac also drove in three runs apiece for the Rays.
Tampa Bay came in hitting a major-league worst .163 and had scored just 20 total runs, the fewest after nine games since the 2003 Detroit Tigers – a team that finished 43-119.
Boston's loss came in Carl Crawford's first game against his former team. Jacoby Ellsbury hit a solo homer for the Red Sox.
Crawford, who spent 12 years in the Rays' organization before leaving to sign a $142-million, seven-year contract with the Red Sox in December, went 2 for 5, raising his average to .163.
Jeremy Hellickson (1-1) gave up two runs, five hits, walked five and struck out one in 5 1/3 innings to earn the win.
Fuld doubled into the left-field corner in his last at-bat in the ninth.
The Rays, who led for just two innings all season, jumped ahead 1-0 on Damon's homer in the first and never trailed.
Damon, a key contributor to Boston's 2004 World Series title team that ended an 86-year championship drought, was greeted with the usual boos he's received since signing with the Yankees following the 2005 season. Before the noise subsided, he belted Daisuke Matsuzaka's first pitch into the Rays' bullpen for his second homer.
It got a lot worse for Matsuzaka (0-2) in the second – and the boos were much louder for him. Tampa Bay sent 10 batters to the plate, scoring six runs on three consecutive two-run hits after the right-hander loaded the bases with no outs. Jaso made it 3-0 with a two-run double, and Brignac drove in a pair with a single before Fuld hooked his two-run shot around the Pesky's Pole in right.
Matsuzaka lasted just two-plus innings, giving up seven runs on eight hits.
Adrian Gonzalez's RBI triple cut it to 6-1 in the third.
The Rays added a run on Damon's RBI single in the fourth before David Ortiz tripled and scored on J.D. Drew's single in the bottom half.
Ortiz became the only AL player with at least one triple in each of the last 12 seasons.
Ben Zobrist's two-run double chased reliever Tim Wakefield and made it 12-2 in the sixth.
Notes
It was Matsuzaka's 100th career start. … Red Sox manager Terry Francona said he'd like to become "somewhat consistent" with his lineup after switching fairly often during the team's initial eight games. … Rays manager Joe Maddon said he hadn't heard from major league baseball after his ejection in Sunday's loss to the Chicago White Sox. He did say he saw highlights of it while he was at dinner Sunday night and said it looked like "an out of control manager" and found the video "somewhat humorous." … Red Sox C Jason Varitek turned 39 Monday. … Damon spoke to a large press gathering before the game. "Those four years coming here as a Yankee, there was definitely a little higher stress," he said. "I feel like I've been a pretty good ambassador to the game. I feel like I'm a pretty good person. A lot of people around these parts dislike me because I did play for the Yankees, that's part of it."