Marc Twaddle had earlier given Falkirk the lead from close range after errors from Perth goalkeeper Graeme Smith.
But top scorer Craig netted just before half-time after Brian McLean was controversially adjudged to have handled before Deuchar claimed the win.
The result takes the Perth side 11 points clear of the bottom-placed Bairns, with a top-six place now a possibility. For Falkirk, their battle to stay up only grows ever tougher.
Scott Arfield was suspended for Steven Pressley's side with Pedro Moutinho coming in.
Saints, making their first visit to the Falkirk Stadium on league business, made three changes from the side that lost to Motherwell. Steven Anderson and Graham Gartland were ruled out, with Kevin Moon on the bench.
In came Gary Irvine, Danny Grainger and Jody Morris for Derek McInnes' men.
With tickets still valid from the original game on January 2, a healthy crowd was in place.
Referee Stevie O'Reilly, who controversially postponed that fixture, was in charge on Saturday.
It was an understandably tentative opening by both sides in blustery conditions on a difficult, bare pitch.
It was eight minutes before the first shot when Moutinho's free-kick was partially cleared by Michael Duberry, but Vitor Lima could only fire over from 16 yards. Deuchar replied with a tame header easily gathered by Robert Olejnik.
It was a tight contest but Falkirk should have done better when Morris gave away possession, allowing Moutinho to break clear only for Lima to slash wide.
But it was not long before Falkirk were handed the opening goal.
Goalkeeper Smith first needlessly raced out to the far side as he attempted to beat Ryan Flynn to the ball.
But Smith did not get enough on his initial challenge and was forced to bring the former Liverpool youth player down, earning himself a yellow card.
Worse soon followed for the keeper as he missed Moutinho's free-kick and Twaddle was on hand to knock home from a yard out on 27 minutes.
The setback fire up the visitors and Steven Milne, Chris Millar and Murray Davidson all had quick headed attempts, the best from Millar which forced Olejnik into a point-blank save.
Flynn dragged wide from 18 yards at the other end.
But it was Saints who grabbed a timely penalty equaliser on the stroke of half-time.
Grainger's high ball to the back post picked out Deuchar whose header back struck McLean's hand, prompting O'Reilly to point to the spot despite strong home protests. Craig neatly tucked home his ninth goal of the season.
McLean almost made up for the penalty concession four minutes after the break. The defender's flick-on from Moutinho's corner looked net-bound until Duberry blocked. In reply, Davidson fired narrowly past with Olejnik sprawling.
Burton O'Brien then wastefully struck a free-kick over from 22 yards.
Lima was not enjoying the best of shooting days and he again passed up an opportunity when Enoch Showunmi picked him out at the back post.
Moutinho was withdrawn for winger Compton 10 minutes from the end, but it was the visitors who took the points with an 84th-minute winner.
Scobbie completely misjudged a long kick forward, allowing Deuchar to lob over Olejnik for a vital away success.
Source: Team Talk