Thistle got in front from the penalty spot, Sean Welsh netting after Ryan Dow had fouled Kris Doolan.
County threatened when Craig Curran fired against the post.
The hosts lost Liam Lindsay to a neck injury in the second half and Welsh threatened again before Chris Burke fired County level.
It had appeared Thistle, who were the better side for much of the game, were about to record their first victory since the opening day of the season.
But, just as they have done recently against Hamilton Academical and Hearts, they conceded a late goal to drop points.
In fact, they could have lost all three points as Ryan Scully saved instinctively from Ryan Dow moments after Burke's equaliser.
Alan Archibald will once again rue missed opportunities and a failure to capitalise on their overall dominance of the match.
Lindsay's serious-looking injury after coming into collision with Scully and Andrew Davies caused disruption to the Thistle backline but the points should already have been safe.
Archibald's managerial acumen - despite Thistle's lowly league position - was recognised by Shrewsbury Town's enquiry this week.
As he sought to put an end to his side's eight-game winless run, he shuffled his pack, dropping Adam Barton into a back three and using Callum Booth and Christie Elliott as wing-backs.
Not only did that provide Thistle with energy and width, but it allowed their central midfield trio of Welsh, Abdul Osman and Ryan Edwards to dominate the County duo of Ian McShane and Martin Woods.
It meant they bossed the game and though Lindsay had to be replaced by Ziggy Gordon, it appeared Thistle had found the means to finally win again.
Media playback is not supported on this device
Just as Thistle's winless run continues, so does County's, though Jim McIntyre will be a lot happier that it does so with a draw rather than another defeat.
They are now without a victory in seven matches, but Burke's injury-time strike at least put an end to their goalless streak.
County scarcely threatened Scully's goal before that but did come within a couple of inches of an equaliser when Craig Curran steered a shot against the post.
But the continuing absence of striker Liam Boyce through injury is hurting the Staggies and McIntyre will hope his return comes sooner rather than later.
Partick Thistle manager Alan Archibald: "We were very good, dominant for most of the match, but County nearly stole it. We need to turn those into victories. That's too many points that have been dropped at home."
Burke was bundled over in the box as County began to press in injury time before eventually grabbing the equaliser through the former Rangers and Scotland winger.
Ross County manager Jim McIntyre: "It's a stonewall penalty. I've watched it back. If the ref deems the first one a penalty, that's every bit as much a penalty. He's got one right and one wrong. He's made a mistake so that's disappointing."
Match ends, Partick Thistle 1, Ross County 1.
Second Half ends, Partick Thistle 1, Ross County 1.
Corner,  Ross County. Conceded by Danny Devine.
Attempt missed. Michael Gardyne (Ross County) right footed shot from outside the box is close, but misses the top right corner.
Attempt saved. Ryan Dow (Ross County) right footed shot from outside the box is saved in the bottom right corner.
Goal!  Partick Thistle 1, Ross County 1. Chris Burke (Ross County) left footed shot from the centre of the box to the bottom right corner.
Attempt missed. Callum Booth (Partick Thistle) right footed shot from outside the box is close, but misses the top left corner from a direct free kick.
Ryan Dow (Ross County) is shown the yellow card for a bad foul.
David Amoo (Partick Thistle) wins a free kick in the attacking half.
Foul by Ryan Dow (Ross County).
Attempt blocked. Chris Burke (Ross County) right footed shot from outside the box is blocked.
Foul by Ryan Edwards (Partick Thistle).
Tony Dingwall (Ross County) wins a free kick on the left wing.
Tony Dingwall (Ross County) wins a free kick on the left wing.
Foul by Callum Booth (Partick Thistle).
Attempt missed. Chris Burke (Ross County) right footed shot from outside the box is high and wide to the left.
Substitution, Partick Thistle. David Amoo replaces Kris Doolan.
Substitution, Ross County. Tony Dingwall replaces Craig Curran.
Attempt missed. Sean Welsh (Partick Thistle) right footed shot from outside the box is too high.
Foul by Christie Elliott (Partick Thistle).
Greg Morrison (Ross County) wins a free kick in the defensive half.
Substitution, Partick Thistle. Ziggy Gordon replaces Liam Lindsay because of an injury.
(Partick Thistle) wins a free kick in the attacking half.
Foul by Andrew Davies (Ross County).
Foul by Ryan Edwards (Partick Thistle).
Chris Burke (Ross County) wins a free kick in the defensive half.
Corner,  Partick Thistle. Conceded by Paul Quinn.
Substitution, Ross County. Chris Burke replaces Marcus Fraser.
Substitution, Ross County. Greg Morrison replaces Alex Schalk.
Foul by Ryan Edwards (Partick Thistle).
Ian McShane (Ross County) wins a free kick on the right wing.
Corner,  Partick Thistle. Conceded by Andrew Davies.
Attempt missed. Danny Devine (Partick Thistle) right footed shot from outside the box is too high.
Corner,  Partick Thistle. Conceded by Paul Quinn.
Adebayo Azeez (Partick Thistle) is shown the yellow card for a bad foul.
Adebayo Azeez (Partick Thistle) has gone down, but that's a dive.
Abdul Osman (Partick Thistle) wins a free kick in the defensive half.
Foul by Ryan Dow (Ross County).
Corner,  Partick Thistle. Conceded by Marcus Fraser.
Attempt blocked. Ryan Edwards (Partick Thistle) right footed shot from outside the box is blocked.
Partick Thistle were pegged back late on by Ross County in the Scottish Premiership.