Chicago took a 2-1 lead in the Eastern Conference quarterfinal series and will try to expand on that advantage when it hosts Game 4 Sunday.
The Bulls led by as many as 17 points in the fourth quarter and Chicago still had a 15-point advantage after Marco Belinelli hit a layup with 5:47 to play.
Noah's free throw came with 4.4 seconds remaining and Chicago survived a missed 3-point attempt by C.J. Watson that would have forced overtime.
Brooklyn made just 34 percent of its field goal attempts, but eventually gave itself a chance to steal the game by holding the Bulls to 14 points in the final period.
Boozer scored 22 points and grabbed 16 rebounds. Luol Deng added 21 points and 10 boards for the Bulls.