The game was decided after midnight and saw each team use five relievers out of the bullpen.
Lucas Luetge (0-2), in his third inning of relief, walked Dustin Pedroia on four pitches to start the 15th and issued an intentional free pass to Mike Napoli after David Ortiz's groundout moved Pedroia into scoring position.
Jarrod Saltalamacchia struck out and Luetge pitched around Jonny Gomes, who saved a potential go-ahead run from scoring in the top half with a sliding catch in left field.
Gomes worked a walk and Drew laced a base hit into the right-field corner to end the 5:03 marathon and give the Red Sox their fourth win in five games.
"Any time you win a game this late in extra innings, it can seem like two," Red Sox manager John Farrell said.
Pedroia finished 2-for-6 with a home run and three RBI, while Kyle Seager led Seattle's 16-hit attack with three hits, including a homer, and two RBI.
Both starters were long gone by the time things settled. John Lackey lasted seven innings for Boston and allowed three runs on eight hits, while Hisashi Iwakuma gave up just two unearned runs on seven hits in 5 2/3 frames.
Drake Britton (1-0) picked up his first big-league win with two scoreless innings in relief.
Seattle hit the scoreboard in the fourth, as Brad Miller hit a leadoff single and scored on a one-out double from Seager.
Boston responded with two in the fifth. Runners were on the corners with none out before Nick Franklin booted a Shane Victorino grounder, allowing Brock Holt to score. Pedroia's fielder's choice grounder plated Jacoby Ellsbury for a 2-1 Red Sox lead.
The M's reclaimed the lead in the sixth on a Kendrys Morales sac fly and two- out RBI single from Raul Ibanez, and the Sox failed to make headway despite having two runners in scoring position with two outs in the home half.
Victorino began the home seventh with a single, and Pedroia launched a ball into the parking lot beyond the Green Monster for a 4-3 Sox edge.
Seager reached the bullpen in right-center with one out in the eighth for a tie game.
Seattle couldn't score with two on in the 11th, then failed to capitalize with runners on the corners in the 12th.
Boston stranded a man on second in the 13th. Brandon Snyder was thrown out at home plate trying to score from third in the 14th.
Both teams struggled with runners in scoring position, with the Red Sox going 1-for-11 prior to Drew's game-winner and the Mariners recording just three hits in 11 chances.
"These guys battled and unfortunately we ended up on the losing side of things," Mariners acting manager Robby Thompson said. "Give it to our guys, they battled for that whole game."
It was the 10th walk-off win of the season for Boston ... Following Tuesday's series-opening 8-2 win, Boston acquired right-handed pitcher Jake Peavy in a three- team deal with the Chicago White Sox and Detroit Tigers. The Red Sox sent minor league pitchers J.B. Wendelken and Francelis Montas and infielder Cleuluis Rondon to the White Sox. Boston also received right-handed pitcher Brayan Villarreal and outfielder Avisail Garcia from the Tigers for shortstop Jose Iglesias. The Red Sox then sent Garcia to the White Sox ... During the game, the Mariners announced the trade of infielder Robert Andino to the Pirates in exchange for a player to be named later or cash considerations.
08/01 02:58:50 ET