A first half penalty from Sean Walsh looked set to be enough to earn the Jags three points but Burke popped up to break the home fans' hearts.
The opening 25 minutes were uninspiring, with chances at a premium.
The match burst into life when the referee pointed to the spot after Thistle striker Kris Doolan was tripped in the box by Ryan Dow. Sean Welsh stepped up to take the spot-kick and calmly slotted it past former Jags goalkeeper Scott Fox to put the hosts 1-0 up.
On the stroke of half-time, Thistle defender Adam Barton came within inches of scoring a goal of the season contender but his long-range effort went just over the bar.
Ross County - who before this game had not scored in over six hours of football - finally got their first clearcut chance of the game. Striker Craig Curran found himself in space in the Thistle box but his shot blasted off the post.
With around 20 minutes to go there was real concern for Partick Thistle as they lost defender Liam Lindsay to a head injury. The 21-year-old received medical treatment for around 10 minutes before being taken off on a stretcher.
Moments after the play resumed, Sean Welsh came close to doubling Thistle’s lead. Ryan Edwards’ quick footwork set up Welsh, but his long range effort went just over the bar.
There were nine minutes of additional time added at the end of the game as a result of Liam Lindsay’s head injury.
In the fifth of these, Thistle were stunned as Chris Burke lashed in a late equaliser. The former Scotland international found himself in space in the box and dispatched a cool finish past Ryan Scully to level the scores.
With just two minutes left, County almost stole all three points but Scully made a fantastic save to prevent an unlikely win.
The frustration grows for Partick Thistle who are now without a win in nine games in all competitions. The Firhill side has now dropped points in the final ten minutes of their last four matches.