The Hatters were gifted the lead in the 18th minute when left-back Ryan Garry sliced the ball into his own net off the post following Lewis Emanuel's cross from the left.
But Molesley, making just his second start since a summer move from non-League Grays, levelled two minutes later, brushing teenager Jake Howells aside and cutting in from the right flank before beating Conrad Logan at his near post.
Josh McQuoid hit the outside of the post from 25 yards in the 23rd minute as the visitors went on to boss the half. The Cherries went ahead six minutes into the second half when a break the length of the field ended with Lee Bradbury feeding Molesley, who turned Logan before sliding the ball into the empty net from 14 yards.
Luton levelled four minutes later when Kevin Gallen rifled a 20-yard free-kick through the arms of Shwan Jalal for his first Luton goal.
Bournemouth struck again in the 76th minute when a quick throw-in from Molesley allowed Bradbury to flick the ball across for the unmarked Matt Tubbs to glance home from six yards. But an enthralling encounter ended all square when Paul McVeigh slammed home from the penalty spot five minutes from time.