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.
Caley made no changes to the side which won 3-0 against Hibs on Saturday, sending them into second in the league. Stuart Kettlewell replaced Gary Glen in the only change to the County side which lost 3-2 to Motherwell on Saturday.
County substitute Grant Munro had a header cleared off the line by David Raven.
McMenamin was then played through, but his low effort rippled the side netting.
County substitute Steven Ross headed a Mark Fitzpatrick cross just wide then visiting captain Richie Brittain had the chance to grasp a last-minute equaliser but it was cleared by Josh Meekings.