Falkirk looked to have the points wrapped up through an early own goal from Harald Pinxten, but Livi battled back well in the second half.
Snodgrass picked up the ball wide on the left and then curled a superb shot past Matt Glennon to send the Livi fans home happy.
Darryl Duffy had an opportunity after only three minutes when the excellent Russell Latapy set him up, but his shot thumped against Roddy McKenzie's right-hand post.
Latapy then attempted to beat McKenzie with a 25-yard free-kick which dipped below the crossbar, but McKenzie managed to flip the ball over the top.
The goalkeeper was badly at fault after 15 minutes when Falkirk were gifted a goal. He missed Craig McPherson's corner and the ball struck the head of defender Pinxten and went into the empty net.
Latapy was proving a thorn in the flesh with his trickery up front and he set up another chance for Duffy, but the angle was tight and McKenzie plucked his chip out of the air.
Snodgrass tested Glennon again early in the second half with a shot from distance which the goalkeeper batted down and gathered at the second attempt.
Duffy had the ball in the net for Falkirk after 49 minutes but was ruled offside from Latapy's first-time pass.
Livingston brought on James McPake for Neil Barrett 10 minutes later and Martin Scott dropped back to midfield, and they carried more threat after that.
Pinxten was not far away with a terrific shot from the edge of the area which beat Glennon but skipped just wide of the post.
It was a clear warning to Falkirk who continued to pass the ball well, but too often wasted the final ball when they closed in on McKenzie's goal.
Latapy set up a great opportunity for Stephen Thomson after 72 minutes, but McKenzie leapt to his left to get a hand to the ball and push it to safety.
The Trinidad & Tobago international was then denied a goal for himself when his 12-yard shot struck the head of Greg Strong just when it looked a certain goal.
But the goal came at the other end as Snodgrass repeated his heroics against Rangers in midweek to snatch a point for Livi.