The 24-year-old has now scored seven goals in his last five games with his brace setting up a last 16 tie with Kilmarnock.
McKay fired home an early penalty before Iain Vigurs' sublime finish levelled things just before half-time, but a spectacular curler from the hosts' in-form striker settled things nine minutes after half-time.
The Highland neighbours met in the aptly named Cold Firm derby after sharing a thrilling 3-3 draw in Dingwall 10 days ago.
Inverness started where they left off against Hibs and took a ninth-minute lead. Former Inverness stalwart Ross Tokely was adjudged by referee Craig Thomson to have taken down Gary Warren in the area and McKay made no mistake from the spot.
The freezing conditions soon saw the game deteriorate into a battle of attrition as both teams had spells on top but failed to carve out anything to threaten either goal. No further chances were created until the 35th minute when Ross Draper found Aaron Doran just outside the area and the Irishman, making his 50th appearance for Inverness, shot inches wide.
It may have been a costly miss as County levelled four minutes before the break. Staggies fans will not have minded waiting for their opportunity with Vigurs netting a beautiful equaliser. The ex-Inverness midfielder dinked the ball over charging goalkeeper Antonio Reguero from 25-yards after he was found by Colin McMenamin.
Draper might have put the home side back in front on the stroke of half-time but he screwed his low piledriver just wide from Richie Foran's lay off. County started brightly after the interval with Rocco Quinn forcing Reguero into a save, but moments later McKay netted his second of the game with a superb curling strike from just outside the box for his 13th goal of the season.
Vigurs had the opportunity to level the scores after 68 minutes when he connected with a Quinn cross but the close-range header was straight at Reguero. Seconds later County substitute Grant Munro watched as his header was cleared off the line by David Raven.
McMenamin was next to try his luck after being played through by Scott Boyd, but his low effort rippled the side netting. County substitute Steven Ross headed a Mark Fitzpatrick cross just wide with seven minutes remaining and visiting captain Richie Brittain had the chance to grasp a last-minute equaliser but it was cleared by Josh Meekings.