Paul Hartley’s men looked on course to rebound from last week’s 6-2 thrashing to city rivals Dundee United when the bottom-of-the-league Staggies had Terry Dunfield sent off early on.
Jim McAlister put the Dees in front just after half-time but Curran equalised to earn the visitors an unlikely point.
The home side started brightly when Paul McGowan ran onto David Clarkson’s pass and chipped Ross County goalkeeper Antonio Reguero but the ball sailed just over.
Disaster struck for Jim McIntyre’s men in just the ninth minute when midfielder Dunfield received his marching orders for serious foul play following a bad tackle on Paul McGowan.
Gary Irvine almost made County play soon after when he latched onto the ball in the visitors’ box but his shot found the side netting.
Filip Kiss came close for the Staggies on 20 minutes when his right-footed shot from the edge of the box was deflected inches wide of the Dundee goal.
The Dees should have been in front just before the half-hour mark when Greg Stewart rolled the ball to Clarkson but the striker somehow miscued his left-footed strike from a few yards out.
The hosts took the lead soon after the break when McAlister ran on to Gary Irvine’s pass and lashed the ball past Reguero with his left foot.
Dundee almost doubled their advantage minutes later when James McPake forced Reguero into a smart save and Luka Tankulic hit the post from the rebound.
The Dark Blues were left to rue that miss when new Staggies signing Curran, who only joined the club this week, smashed Kiss’ cross into the bottom-left corner on 65 minutes to level the game.
Ross County were forced to hang on in the final minutes as Dundee frantically pressed for a winner.
McAlister forced a good save from Reguero after getting on the end of Mark Stewart’s knockdown before Kevin Thomson shot narrowly wide from the edge of the box.
Irvine came close for Dundee again at the death but Reguero produced an excellent save to deny him and earn his side a valuable point which lifts them level with St Mirren at the bottom the Scottish Premiership on 11 points while Dundee remain eighth with 22 points.