The 76ers knotted the Eastern Conference semifinals at 2-2 with Game 5 scheduled for Monday in Boston.
The Celtics appeared in control when they took a 15-point lead at halftime, but Philadelphia came back thanks to 19 offensive rebounds and then dominated down the stretch.
Boston forged an 83-83 tie with 1:38 to play on a layup by Rajon Rondo, but the Celtics could not produce another point.
Iguodala started the late charge with a jumper that gave Philadelphia the lead for good with 1:22 left and, after Rondo had a layup attempt blocked by Thaddeus Young, Iguodala hit a 3-pointer that put the 76ers in front by five.
Four free throws by Jodie Meeks over the final 32 seconds finished off the scoring.
Boston managed just 37 points in the second half and eventually lost the game due to a 52-38 rebounding deficit.
Philadelphia's bench players outscored their Boston counterparts 43-12.