Milwaukee made up an eight-point deficit at the start of the fourth quarter to take a five-point advantage with 33 seconds to go in regulation, but Boston came back to force overtime when Paul Pierce hit a 3-pointer with 2 seconds remaining.
The game was knotted at 94-94 in overtime when Ellis made 1-of-2 free throws. Brandon Jennings added two foul shots for the Bucks with 19 seconds left and, after Pierce missed a 3-point try, Larry Sanders made a dunk to clinch the victory.
Boston suffered its fourth loss in five games, but this was the Celtics' first home setback in December.
Ellis finished with 27 points, Luc Richard Mbah a Moute had 20 points and Sanders delivered 17 points and 20 rebounds.
The Bucks overcame a 35-point performance from Pierce, who also pulled down 12 rebounds.