Brian Graham’s first-half opener looked to have secured all three points for the visitors until Murdoch capped an improved second half from the Bairns with a deserved equaliser.
After a scrappy opening devoid of clear-cut chances, Jason Thomson and Joe Hamill both fired over the bar before Rovers opened the scoring on the half hour with the first effort on target.
Thomson got in behind the home defence and drilled over a low cross with Graham converted from close range.
A fierce half-volley from Hamill shaved the post, and Jay Fulton’s snapshot from the edge of the box was saved by David McGurn shortly after the break.
McGurn again denied the lively Fulton and produced another fine stop to keep out a powerful header from Laurie Ellis before Johnny Flynn nodded over.
But Falkirk’s pressure was finally rewarded with seconds remaining as Murdoch climbed well to power Thomas Grant’s corner past McGurn.