The Rugby Park outfit dominated the Ladbrokes Premiership encounter, but found County goalkeeper Ross Laidlaw in inspired form as he kept them at bay.
Laidlaw pulled off a string of impressive saves, but the fact his one to keep out Sow's second-half spot-kick was not top of the list was an indication of how poor the forward's effort was.
The goalless draw meant Kilmarnock missed out on the opportunity to move into the top six, with County occupying fifth place after their impressive start to life back in the top flight.
County were first to threaten when Michael Gardyne created space but then drilled his shot over the crossbar.
And the visitors had a strong appeal for a penalty when Ewan Henderson's well-struck shot looked to have been blocked by Alex Bruce's outstretched arm as the Kilmarnock defender rushed out to charge down the shot, but referee Alan Newlands was unmoved.
Kilmarnock gradually began to gain the upper hand as the first half wore on and Bruce's glancing header from a Rory McKenzie free-kick was turned away by Laidlaw.
The keeper then made an impressive stop to keep out Sow's powerful 25-yard drive.
McKenzie came closest to breaking the deadlock in the first half when Stephen O'Donnell slid a through-ball in behind the County defence for the winger and he unleashed a shot that rattled a post.
McKenzie was involved again when he latched on to a cross from O'Donnell and rolled the ball into the path of Sow, but the striker was unable to take it in his stride as he went towards goal and the chance was lost.
Kilmarnock continued to push for the opener after the interval, with left-back Niko Hamalainen testing Laidlaw.
The keeper then pulled off an impressive stop to claw away a Bruce header after Sow had headed a McKenzie cross back into the six-yard box.
Laidlaw was proving to be an invaluable last line of defence as he turned away another Sow drive from outside the area.
Kilmarnock were presented with a golden opportunity to break the deadlock in the 64th minute when County captain Marcus Fraser's clumsy challenge on Liam Millar on the bye-line resulted in the referee pointing to the spot.
Sow stepped up to take the penalty, but the striker fluffed his lines with a weak spot-kick that was comfortably saved by Laidlaw.
At the other end, Laurentiu Branescu had been a virtual spectator for the majority of the match but was called into action moments after Sow's penalty miss when Joe Chalmers curled a free-kick towards the top corner, the Kilmarnock goalkeeper getting across his line to claw the ball away and ensure the match finishes goalless.