"We had guys come to spring training, everybody bought in," Breslow added, explaining Boston's quick turnaround after going 69-93 last year. "There's accountability and 25 guys who prioritize winning baseball games beyond any kind of individual achievement or accolade."
The resilient Rays won four win-or-go-home games over the previous nine, but couldn't win another to send the best-of-five matchup back to Fenway Park for a decisive Game 5.
Shane Victorino's infield single snapped a seventh-inning tie and Dustin Pedroia's ninth-inning sacrifice fly provided insurance for the Red Sox.
Koji Uehara got the final four outs one night after giving up a game-winning homer and Boston rebounded to take the series in four games.
The Red Sox will open the ALCS at home Saturday against the Athletics or Tigers. Oakland hosts Detroit in a winner-take-all Game 5 on Thursday.
There was a sense of relief in finally getting past the Rays, who have been a tough matchup for Boston over the past six seasons. Counting three wins in the ALDS, the Red Sox won 15 of 23 meetings between the teams this season.
"I'm assuming that the next opponent we have is going to be as tall a challenge as Tampa is," manager John Farrell said.
"You never feel like you're comfortable going against them. We've had some success against them this year, where maybe in years past it's been a little bit different," he added. "They posed a stern challenge for us, no question."
Both managers mixed and matched all night in a tense game that felt more like a chess match. Desperately trying to force a fifth game, Rays skipper Joe Maddon used nine pitchers a postseason record for a nine-inning game and had ace David Price warming up for a potential 10th inning.
"The way it was working at the beginning there, I could see it was just not going to work and we had to do something differently," Maddon said. "We became a little bit more extemporaneous at that point."
Breslow relieved Boston starter Jake Peavy in the sixth and struck out his first four batters all in the middle of Tampa Bay's lineup. The 33-year-old lefty from Yale has pitched for six teams in eight big league seasons, including two stints with the Red Sox.
The highest-scoring team in the majors this season, Boston scratched out three runs on six singles in a game that featured only one extra-base hit.
"They didn't make any mistakes. You could see their grit," Maddon said. "They've got a bunch of gamers over there. ... On the other side, I think our guys were equally as tough. We have had a hard time hitting their pitching staff."
Making their fourth playoff appearance in six years despite having one of the lowest payrolls in baseball, the budget-minded Rays have not advanced past the division series since reaching the 2008 World Series.
Xander Bogaerts scored the tying run on Joel Peralta's wild pitch in the seventh and Victorino followed with an RBI infield single. Pedroia drove in Bogaerts with a sacrifice fly in the ninth to make it 3-1, and Uehara struck out Evan Longoria to end it.
"It feels outstanding. This is what you play for," said Peavy, who allowed one run and five hits in 5 2-3 innings. "That Tampa Bay team is outstanding. Hats off to them. They are an outstanding baseball team well-coached and they played hard. We knew each other very well, so it's tough. I can't tell you how happy I am and the rest of these guys."
When it was over, the Red Sox mobbed Uehara in front of the mound, then took the party inside to the clubhouse, where music blared and champagne flowed.
The Rays retreated to a solemn clubhouse.
"Obviously, everybody is disappointed right now. Joe (Maddon) came in here and said to not hang our heads and remember that we had our backs against the wall several times and played really well to get to this point," second baseman Ben Zobrist said. "It was a good year overall, but it's tough to take the loss."