Antetokounmpo and Middleton were the first pair of Bucks teammates to score 30 apiece in a playoff game since Bob Dandridge and Brian Winters did so in 1976.
Terry Rozier, starting in place of injured Kyrie Irving, added 23 points, Marcus Morris scored 21 off the bench while Jaylen Brown netted 20 and Jayson Tatum contributed 19 points and 10 rebounds for the Celtics.
Stevens said he plans to have a lot of different people on him, but the main man will be Al Horford, as he said: "We have to throw a lot of different bodies at him".
With the game tied at 96 after the Bucks' Malcolm Brogdon made a three-pointer, the Celtics had a chance to take the lead.
Middleton scored 31 points, while Antetokounmpo had 29 points and 11 rebounds for the Bucks.
Milwaukee and Boston split their four-game, regular-season series, with each team winning once on the other's home court.More news: Russian Federation calls an emergency session to discuss strikes on Syria
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"I know the IN game when I first hurt it, it felt like there was a knot IN the ball", Smart said.
The three-pointer came just seconds after Celtics reserve guard Terry Rozier hit a massive step-back three that seemed to put the game away.
Antetokounmpo made one of two free throws at the 14.8 mark, and then committed his sixth and final foul fighting for a loose ball with Rozier, who made one of two free throws with 13 ticks to go.
An acrobatic, spinning layup by Tatum gave Boston a 108-105 lead in overtime.
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.
Middleton hit a desperation 3-pointer from near mid-court as time expired to send it to overtime. With a splint on his shooting hand, the 24-year-old took part in shooting drills on Sunday ahead of Game 1, and sounded optimistic that he'll be back on the hardwood competing sooner rather than later. "So the thumb felt well", said Smart on Sunday, according to ESPN's Chris Forsberg.