The Scottish Championship side needed a replay to knock Killie out of the competition.
A stoppage-time goal from Nicky Clark ensured that the Light Blues progressed to the last eight at the expense of Kilmarnock, and McKay explained that while they were pleased to knock a top flight side out of the competition, it did little for their belief as they already felt confident that they can go far in the cup.
"All of the boys believe we can win the Scottish Cup this year and that is our intentions," he told Sky Sports. "We believe we can do it and now it's about showing we can get there.
"It wasn't vital that we beat a Premier League side to show we could do it, we approach all of our games like this as we do in every game. It's about going to show what we can do and we managed to do that against Killie which was great."
While the Gers have a home tie to look forward to in the next round, they will know that they may have to see off Premiership opposition once again if they hope to make it to the semi-final. Dumbarton and Dundee could not be separated when they met earlier in the month, so will meet again next Tuesday to decide who travels to Ibrox to face Mark Warburton's men.
And while the prospect of facing Championship rivals Dumbarton is unlikely to concern the Rangers squad too much - as they have won all of their previous meetings between the pair this season, with the last two games finishing 4-0 and 6-0 to the Light Blues - Dundee should be a tougher test than the club's previous top flight opponents.
Paul Hartley's men are unbeaten in their last four games and sit in the top half of the Premiership table. And with Celtic, Ross County and Inverness all still in the competition, it appears that the Gers will need all of the belief that McKay speaks of if they hope to be lifting the trophy for the 34th time.