Simon Grayson's men were looking to move into the play-off places by claiming all three points at Turf Moor, but their quest looked set to prove unsuccessful as the Clarets dominated for large periods.
Eddie Howe's side were set on their way early on as Jay Rodriguez nodded home.
But despite being under the cosh for much of the clash, the visitors equalised courtesy of Snodgrass's third goal of the season.
And the Scotland international added another in the 89th minute to ensure Leeds made it back-to-back victories.
In contrast, Burnley's third successive loss leaves Howe's position increasingly precarious.
Seemingly refreshed by the recent international break, the visitors started positively and fashioned the first opening when Adam Clayton volleyed just over.
However, their bright start did not last long as the Clarets took the lead in just the tenth minute.
Kieran Trippier was the orchestrator, sending in a teasing cross which Rodriguez nodded past a helpless Alex McCarthy.
The home side pushed for a second and almost got it moments later when Keith Treacy cleverly worked an opening before seeing McCarthy keep out his toe-poke.
But they suffered a blow in the 29th minute when Charlie Austin was forced off through a shoulder injury, handing a debut to Sam Vokes, who joined on loan from Wolves on Friday.
Ross McCormack twice tested Burnley goalkeeper Lee Grant with long-range free-kicks later in the half, but that was all Leeds could muster.
Indeed, the Clarets should have gone into the break two up, but Rodriguez shot straight at McCarthy after getting in behind the backline.
The breathless finish to the half continued as Treacy's sensational 25-yard low strike was kept out by the increasingly busy McCarthy.
Treacy was on hand to test McCarthy once more just after the interval, but the goalkeeper was equal to his 20-yard free-kick.
Treacy must have been starting to wonder what he had to do to score midway through the half, this time being thwarted by the crossbar after producing another thunderous effort.
The home side were left rueing their inability to grab a second in the 75th minute as Lloyd Sam's cross picked out Snodgrass, who slid home at the back post.
Burnley continued to dominate thereafter, with Rodriguez firing just over late on.
And that was to prove costly as, with two minutes remaining, the ball broke to Snodgrass to chest it down at the back post and finish coolly.