Saints suffered an early blow after only 11 minutes when Steven Robb was stretchered off injured and replaced by Garry Brady.
Another setback might have followed almost immediately but Falkirk defender Lee Bullen directed a free header over the bar from close range when he seemed certain to score.
The home side capitalised on their lucky escape in the 17th minute when Dorman managed to slip away from his marker before hammering a fierce right-foot drive under the body of goalkeeper Dan Mallo.
Falkirk were clearly rattled and Mallo was called upon to make an important reflex stop from Mo Camara's sweetly struck free-kick from fully 30 yards.
The Bairns then grabbed an equaliser out of the blue as Carl Finnigan turned quickly to guide an angled right-foot effort across goalkeeper Chris Smith and into the far corner.
Home midfielder Hugh Murray and Falkirk striker Michael Higdon both headed over the bar shortly before half-time and the game continued to go from end-to-end in the second period with both sides refusing to settle for a point.
Saints had more of the ball as the visitors began to sit deeper as the contest wore on and substitute Brady saw a right-foot drive flash just the wrong side of the post after 63 minutes.
Falkirk counter-attacked quickly and from an inch-perfect Finnigan cross, substitute Mark Stewart headed straight at Smith.
Moments later, Buddies top scorer Billy Mehmet shot fractionally wide but man of the match Dorman made no mistake with a decisive finish from eight yards with only 12 minutes remaining.
Falkirk appeared beaten yet they pulled level for the second time with five minutes to spare when Jack Ross handled a header from substitute Kevin Moffat, allowing Scott Arfield to send Smith the wrong way from the penalty spot for his seventh goal of the season and perhaps the most crucial so far.