Odds have tumbled on the League One Ibrox men becoming the first team from the Scottish third tier to lift the cup.
But McCoist claims that after beating holders Celtic at Parkhead last Saturday, it is Derek McInnes' men who are the safer wager.
Rangers will take on minnows Albion Rovers at home in the quarter-finals on the second weekend in March, while the Pittodrie side host Dumbarton.
The Rangers boss said: "I wouldn't have thought that we'd be the favourites, certainly with the way Aberdeen played at the weekend. Let's be honest about it, most people would install them as favourites and rightly so.
"Dundee United will be a lot of folk's favourites as well.
"We are just delighted to have a home tie in the last eight and hope to give the fans something to look forward to."
Former Rangers chief executive Charles Green angered McCoist at the start of the season when, during a short-lived spell as a consultant to the club, he claimed the manager would need to secure a cup win to save his job.
Despite taking a step closer to that feat, the club's all-time record scorer stuck to his guns as he said that would not be necessary.
"The pressure is not on me from the person who said it, thankfully," McCoist said. "But there is pressure to win trophies here all the time. I personally didn't agree with that statement and I still don't, but you'd expect me to say that."
Aberdeen's victory at Celtic eliminated the possibility of an Old Firm clash this season.
McCoist believes his team could beat their old rivals over 90 minutes but admitted he was slightly relieved not to have to put that claim to test.
McCoist said: "A one-off game is a one-off game and anyone can win that. But to do it across a league, we would absolutely need further investment to compete regularly at the top of Scottish football.
"A game against Celtic would bring its own strains and pressures. I'll tell you the truth - I wasn't particularly wanting to get them and I wasn't particularly wanting to avoid them.
"There is no point worrying about things you can't have an effect on, and unless I was making the draw I can't have an effect on it, so what will be will be.
"It's now clear we won't now be playing Celtic so we will just move on."
Rangers face a trip to third-placed Ayr United - a side they lead by 31 points - on league duty on Saturday.
Friday's second anniversary of the club entering administration had the manager in a reflective mood.
While the Light Blues are overwhelming favourites to win promotion to the Championship, there are lingering concerns about Rangers' financial state after former finance director Brian Stockbridge claimed they could be down to their last £1million by April.
McCoist, who was due to meet Rangers chief executive Graham Wallace on Friday afternoon, said: "It's been two really dark, bleak years for the club that nobody would ever have predicted.
"There have been positives but it goes without saying that the negatives far outweigh those over these past two years. But there are definitely positives for the fans to look forward to.
"We won the title last year by a great number of points and are in a fantastic position to do the same this year. We also have the Ramsdens Cup final and a Scottish Cup quarter-final to look forward to.
"I haven't had any update on the financial situation from the club but I meet with Graham on a regular basis and I do think that if the money was due to run out he would certainly have said to me. But I have had no indication of that at all."