The result means that McCoist is yet to win any of the eight cup competitions that he has led Rangers into as manager, but he believes that the whole club should take blame for the disappointment.
"I don't fear for my future at all. I know the fans won't be happy but no-one will have to tell me anything, that's for sure," McCoist is quoted as saying by PA.
"I've had some big disappointments. That's up there with all the major disappointments that I've had, but it's not just about me - it's about the team and the club and the players, all of us together. So it's a big disappointment and we need to take it on the chin.
"We don't have time to feel sorry for ourselves. That's the one thing we can't do. We've got a big game to look forward to on Saturday."
Rangers do still have one other opportunity to pick up a cup this season as they face Dundee United in the Scottish Cup semi-final next weekend.