"Come in and do what he did, you only see that in the movies," Red Sox designated hitter David Ortiz said. "That guy is a good hitter. One day he's going to get plenty of at-bats. He reminds me of myself when I first started. Don't get to play too much, but with a good swing."
Boston has a season-high 9½-game advantage over second-place Tampa Bay in the division race.
Carp sent a drive over the center-field fence on the first pitch from Roberto Hernandez. It was the first pinch-hit grand slam for Boston since Kevin Millar hit one at Milwaukee on June 7, 2003.
"He hung a slider and I did a lot more than I expected to do with it," Carp said.
Carp is 5 for 17 with two homers and nine RBIs as a pinch hitter this year. Boston has seven pinch-hit homers overall on the season.
"Mike has been so productive in that role," Red Sox manager John Farrell said. "To sit there for nearly four hours and come up and swing at the first pitch you see for a grand slam, that's pretty remarkable."
Dustin Pedroia opened the 10th with a walk against Joel Peralta (2-7) and went to second on Shane Victorino's bunt. After Ortiz was intentionally walked, Hernandez replaced Peralta and walked Mike Napoli on four pitches before Carp went deep for his ninth homer.
Boston closer Koji Uehara (4-0) struck out two during a perfect ninth. He has retired his last 34 batters.
James Loney homered for the Rays, who have has lost 13 of 17. Tampa Bay's lead for the second AL wild-card spot was trimmed to one game over the New York Yankees, who won 5-4 at Baltimore.
The Rays trailed 3-1 before Evan Longoria doubled in a run in the seventh and Loney hit a tying solo shot off Brandon Workman in the eighth.
Napoli had a two-run double and Daniel Nava hit an RBI grounder off Alex Cobb to put the Red Sox up 3-0 in the third. Napoli has driven in 11 runs over his last seven games.
The Rays got one run back in the third on David DeJesus' RBI double. Red Sox shortstop Stephen Drew made a strong defensive play later in the inning with two outs and the bases loaded when he drove to his right to field Desmond Jennings' grounder and then throw out Matt Joyce at second base.
"An awesome play ... what a play," Red Sox starter Ryan Dempster said.
Dempster, who gave up one run, four hits and five walks in five innings, struck out Jennings to leave the bases loaded in the fifth.
"You can slice and dice it, but we had chances to score way more than three runs tonight and didn't do that," Rays manager Joe Maddon said.
Cobb allowed three runs and seven hits over 5 2/3 innings.
Tampa Bay just missed turning a triple play in the sixth. With runners on first and second, Longoria leaped to grab Will Middlebrooks' high-hopper near the line and then landed on the third-base bag. Longoria threw out Drew at second, but Middlebrooks just beat the ensuing throw to first.
Farrell said RHP Clay Buchholz came through his first start in three months Tuesday night in good shape and will pitch again on Sunday. Buchholz, who had been sidelined by a strained neck, threw five innings in Boston's 2-0 win over the Rays. ... Tampa Bay is hitless in its last 12 at-bats with the bases loaded. ... Texas Rangers senior adviser Tom Giordano scouted the game on the 60th anniversary of his debut as a major league player with the Philadelphia Athletics, a game in which he hit a home run.
Copyright by STATS LLC and The Associated Press
ST. PETERSBURG, FL - SEPTEMBER 11: Outfielder Shane Vixctorino #18 of the Boston Red Sox questions umpire Vic...
19,215 (56.4% full) - % is based on regular season capacity
Home Plate - Vic Carapazza, First Base - Lance Barksdale, Second Base - Gary Cederstrom, Third Base - Angel Hernandez
Ryan Dempster throws 34 percent of his pitches in the upper-third of the plate, the fourth-highest rate in the AL. The Rays rank in the top five in the AL in slugging and average in at-bats ending with a pitch in that location.