Anthony connected on 13-of-29 shots from the field, including a lay-in off a Boston turnover that put the Knicks ahead by five points with 2 1/2 minutes remaining.
Meanwhile, New York unleashed a harassing defense that limited the Celtics to just eight fourth-quarter points en route to a win in the first contest of their Eastern Conference playoff series.
New York was swept by the Celtics in the first round of the 2010-11 post-season.
J.R. Smith (15 points) and Raymond Felton (13, six assists) also reached double figures.
Swim Week Miami Beach 2014 [PHOTOS]