Rookie Derrick Rose produced an outstanding display for the Bulls, who finished five places behind the Celtics in the East, with 36 points and 11 assists in his first playoff.
Chicago, who had a 41-41 regular season record, led for long stretches and had an 11-point lead in the third quarter.
But Boston, without key power forward Kevin Garnett through a knee injury, fought back to close the third 72-71 up.
Deep in the fourth quarter, Rajon Rondo showed great mobility after picking up a rebound to maneuver for a shot that put the Celtics ahead.
Moments later, though, a Rondo foul gave the Bulls a penalty with less then 10 seconds on the clock to get the two points and regain the advantage.
The Celtics should have sealed the game in the final seconds of regular time when Joakim Noah fouled but Paul Pierce missed the second free throw to take the game into overtime.
Tyros Thomas produced a jump shot with about a minute of overtime left to put the Bulls in front and then the Celtics' Ray Allen, who had a disappointing game, missed the chance to tie the score from the baseline at the end of overtime.
Rose tied Kareem-Abdul Jabbar's NBA record for most points scored by a player in their first playoff game—and broke his own NBA career-best scoring record.
Team mate Ben Gordon said he was not surprised by the way Rose coped with the pressure on his playoff debut.
"Somebody asked me before about him having playoff experience and I said he doesn't need it," said Gordon.
"He's poised beyond his years. He already carries himself like a veteran out there. He had a phenomenal game to say the least," added the London-born Gordon.
Bulls coach Vinny Del Negro praised Rose's "quiet confidence" but said his team would need to improve to get a repeat win in Monday's second match.
"I was pleased with the way they played but we know we are going to have to play better. The Celtics are world champions for a reason," he said.
"I have a tremendous amount of respect for (coach) Doc Rivers and what he has done here so we know that we are going to have to raise our level on Monday.
"Nothing's easy. It's the first game of a long series, and I'm just pleased with the way we hung in there and made plays when we had to," added Del Negro.
Rivers said there had been very little to choose between the teams.
"The playoffs have begun and we're down 0-1. We've put ourselves in a hole.
"Honestly, I thought we did it as much as they did it. But give them credit. Rose was absolutely fantastic."
