The 38-year-old, nicknamed Ball-Run, had trailed 9-8 overnight, but managed to win six of eight frames on Monday afternoon to move 14-11 ahead heading into the final session.
However, Murphy fought back with breaks of 75 and 64 as the 2005 champion took three frames in a row to level at 15-15.
Both players then traded snookers in a marathon 31st frame, which went on for 64 minutes before Bingham finally won 80-56, after collecting 38 points in fouls from his opponent.
After edging back ahead, the world number 10 then took the next frame to move within one of his first Crucible triumph.
Bingham, who beat five-time winner Ronnie O'Sullivan and Judd Trump to reach the final, was visibly emotional as he finished with a break of 88 to become the third-oldest world champion in tournament history.