BOSTON -- Boston coach Brad Stevens loves the postseason because it offers a chance to "bust narratives."
The Celtics took their first step toward shattering the belief they can't win without their biggest stars.
The Celtics led 99-96 with 0.5 seconds left in regulation when Middleton took an inbounds pass on the far right wing and hit a contested 35-footer. Boston outscored Milwaukee 14-8 in the extra period.
Rozier said he knows he's under a spotlight this postseason with Kyrie Irving out for the season following his recent knee surgery.
"I just know I gotta step up," he said. "I know I gotta fill big shoes. I don't feel no pressure. I'm glad to be in this position."
Game 2 in the best-of-seven series is Tuesday night in Boston.
Giannis Antetokounmpo led the Bucks with 35 points, 13 rebounds and seven assists before fouling out in overtime. Middleton had 31 points, eight rebounds and six assists. Malcolm Brogdon scored all 16 of his points after halftime.
After a muted start, things opened for Antetokounmpo. But he said he must do a better job the defensive end against Horford going forward.
"He knows what he's doing down there," Antetokounmpo said. "Hopefully in Game 2 I won't play on my heels and I'll play more on my toes and be able to defend without fouling."
Part of the reason the Bucks were in bad spots is because they were hampered by turnovers. They committed 20, leading to 27 Boston points.
"That's not getting shot up," Middleton said. "We feel like if we get a shot up, we'll be good."
An acrobatic, spinning layup by Tatum gave Boston a 108-105 lead in overtime.
It was 108-106 when Horford snared a rebound and got it ahead to Rozier, who was fouled. He made two free throws to put the Celtics in front 110-106 with 18.8 remaining.
Antetokounmpo made one of two foul shots with 14.8 seconds left, but he fouled out of the game while going for the rebound on his miss.
Rozier added three free throws to provide the final margin.
The game went to extra time following a wild finish in regulation that included back-to-back 3-pointers in the closing seconds.
Milwaukee was held to just two field goals over the first seven minutes of the fourth quarter as the Celtics built an 86-76 advantage.
The Bucks closed to 89-87 before Marcus Morris made a falling down jumper just inside the arc to make it 91-87.
It was 92-91 with less than a minute to play when a pair of free throws by Horford made it 94-91.
Antetokounmpo got free for a dunk. But Horford was steady again after being fouled, connecting on two more from the line with 15.2 seconds left.
Out of a timeout, Brogdon got free on the wing and drained a 3 with 10.3 to play to tie it at 96.
Rozier dribbled the clock down before crossing over Eric Bledsoe, stepping back and swishing an apparent game-winner with a half-second showing the clock.
But after a Milwaukee timeout advanced the ball to half court, Middleton matched it with his own 3 over Brown as time expired.
"Both teams are going to keep fighting," Bucks coach Joe Prunty said. "That's what this series is going to be. Each little play is going to matter."
Bucks: Antetokounmpo has now scored at least 20 points in eight consecutive games against the Celtics. ... Outscored the Celtics 58-44 in the paint.
Celtics: Improved to 23-20 in postseason overtime games. Except for Horford, who entered the Sunday's game with 91 career playoff starts, Boston's other four starters (Tatum, Aron Baynes, Brown, Rozier) had combined to play only one playoff game between them. That lone start belonged to Baynes.
"That was part of our plan, to kind of lull them into a false sense of security." -- Stevens, joking on Boston's play during Bucks' 26-4 run in the second quarter to take a 10-point lead
Tatum brought the crowd to its feet early in the first quarter when he took an outlet pass from Horford and went in for fast-break dunk over the outstretched arm of Antetokounmpo .
Marcus Smart, who hasn't played since March 11 and continues to work his way back from his March 16 surgery to repair a torn ligament, was also on the court getting up some shots before the game. He was recently cleared for non-contact shooting but hasn't resumed full basketball activities.
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