Boston moved to within one victory of the NBA Finals as they beat Miami 94-90, with Kevin Garnett scoring 26 points and grabbing 11 rebounds.
The Celtics took a 3-2 lead in the the best-of-seven Eastern Conference final and will advance with a home win in game six on Thursday.
"Just because we're going to Boston doesn't mean they are going to give it to us," Celtics coach Doc Rivers said. "We're going to have to play. We're going to have to take it."
A seventh game, if needed, would be played at Miami on Saturday. The series winner will play the Western Conference champion, either Oklahoma City or San Antonio, in the NBA Finals starting next Tuesday.
"We're in the biggest challenge of our lives right now, of our professional careers, but that's how it's supposed to be," said Miami's Chris Bosh, who returned to the lineup after being sidelined with an abdominal muscle strain.
"We know what we have to do. There's no reason to dwell on it. Collectively we have to be mentally strong and focus on what it takes to get it done."
The Heat reached the NBA Finals last year, when they lost to the Dallas Mavericks.
But they were pushed to the brink of elimination a stage earlier this time, despite NBA Most Valuable Player LeBron James scoring 30 points and grabbing 13 rebounds, with Dwyane Wade adding 27 points.
"This is what the playoffs are about. It tests your collective resolve," Heat coach Erik Spoelstra said. Boston, he said, "have championship DNA. They have what we're trying to get. A way to get it is with a big Dwyane Wade adding 27 points.in on Thursday.
"It's like a boxer -- we have to go and huddle up in our corner right now. We've shown some resolve and character and we will have to show it again Thursday. We have a pretty tough-minded group. We will be ready."
Miami led 42-40 at half-time and the teams were deadlocked at 50-50 before the Heat scored nine points in a row, only to have Boston end the third quarter with a 15-1 run that put them ahead 65-60 entering the fourth quarter.
"We're just hanging in there," Rivers said. "We just hung around enough to get to the fourth quarter. Our execution down the stretch... I asked a lot of our guys, honestly too much at times, and they came through."
Paul Pierce made two free throws and, after a free throw by Miami's Udonis Haslem, sank a 3-pointer while being guarded by James to give Boston a 90-86 advantage.
"We gave ourselves a chance to win and we were able to do it," Pierce said.
The Celtics' Garnett played a pivotal role late in the game with little rest.
After Wade and Ray Allen traded two free throws each, James scored on a layup to pull Miami closer, but Haslem fouled Garnett and the Boston big man sank two clutch free throws to give the Celtics a 94-90 lead.
Wade had a shot for a three-pointer but missed and Garnett grabbed the rebound, hurling a pass to the other end of the court as the final buzzer sounded.
"He's our life," Rivers said of Garnett. "He does so much that just doesn't have numbers to it. In a strange way, he is calming our guys. He has just been terrific for us.
"We have a lot of positive talk in our locker room," Rivers added. "They believe they can win. We have to just keep hanging in there."
Miami's Bosh, who suffered his injury in the first game of a second-round series against Indiana, scored nine points in 14 minutes but did not play in the decisive fourth quarter.
"I didn't think it would necessarily be fair to throw him in with three minutes to go," Spoelstra said. "He felt good after the game. He might be able to play a handful more minutes."
Bosh said he was fit to play had he been called upon.
"I was ready. If he felt that it wasn't fair, that's great," Bosh said. "I definitely had more to give. It felt good to be back out there. I definitely will be ready to play more (on Thursday) than I played tonight."