Milwaukee's Khris Middleton also torched the Celtics to the tune of 31 points.
But the Bucks got little else from the rest of their team, a key to Boston's 113-107 overtime win to go up 1-0 in the best-of-seven series.
Antetokounmpo and Middleton combined for 66 points, or 61.7 percent of Milwaukee's offense.
Despite their scoring, Celtics head coach Brad Stevens was for the most part pleased with how his team defended Milwaukee's 1-2 scoring punch.
"I thought we did an incredible job on Giannis," Stevens said. "And so he had 35 points, but I thought our bigs kept him in front, made it as tough as possible. He had multiple possessions where he had to stop, pivot, pump-fake, and all that stuff. And still find a way to get the ball up on the glass."
When it comes to Middleton, "he's a really, really smooth shootin' guy," said Stevens, adding, "You know, he averages 20 (points) a game. People don't talk about him enough, probably."
As far as containing the rest of the Bucks who shot a combined 17-for-42 (40.4 percent) compared to the 23-for-41 (56.1 percent) by Antetokounmpo and Middleton, Stevens said there was no specific plan of attack to take away or limit the rest of the Milwaukee players.
"I mean, you still … you just prepare for each guy; Middleton, Giannis, and everybody else for what they do best," Stevens said.
Here are five takeaways from Boston's 113-107 overtime win over Milwaukee in Game 1 of their best-of-seven first round series against the Bucks.
The Celtics could not have been more transparent about their plan to get the ball to Horford a lot when guarded by Antetokounmpo.
Fifteen seconds into the game and Horford had his first basket, a five-foot hook shot over Antetokounmpo in the paint.
Horford scored a team-high 24 points on a ridiculously efficient 5-for-8 shooting.
Of those eight shots, only one was uncontested (a put-back dunk).
Out of the remaining seven field goal attempts, four came when Horford was being defended by Antetokounmpo.
"We wanted to make him play in the post, and make Giannis defend down there," Stevens acknowledged.
Horford Livin' Large at the Free Throw Line
That steady dose of feeding Horford in the post led to the five-time all-star taking a playoff career-high 14 free throw attempts with 13 makes, which nearly doubled his previous playoff career high for free throws made (7) and attempted (8).
"I wasn't really trying to (draw fouls). I was trying to score," Horford said. "But they were fouling me, reaching, whatever. I was trying to be aggressive, score the ball."
Tatum joins exclusive club
Jayson Tatum has been etching his name in the Celtics record book for rookies most of this season. But his latest entry may be the most impressive of them all. He had a double-double of 19 points and 10 rebounds, become just the third Celtic all-time to tally a double-double in his first playoff game. The two other two? Hall of Famers Tommy Heinsohn and Bill Russell.
Second-quarter struggles continue
Boston had yet another woeful second quarter, missing 16 of their 20 shots from the field in the second. And yet the deficit at halftime was just three points (47-44). "They turned it up on us in the second quarter and we were playing well up to that point but not well enough," said Celtics head coach Brad Stevens.
Celtics mental toughness challenged
After seemingly having Game 1 won, only to find themselves heading to overtime following Khris Middleton's 3-pointer that forced overtime, the Celtics were cool as can be in the unexpected extra period.
"I thought it was over," said Boston's Al Horford, referring to the 3-pointer made by Terry Rozier that put the Celtics ahead with 0.5 seconds to play. "Everybody always says play to the last second and everything, but half a second I figured we were good. As soon as he shot it I was like, ‘oh that's good,' I just had a feeling. He hit it and we had to refocus and our guys we did that and just kept on grinding it out."