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The Parkhead hero admits that if the two sides were to meet again right now, he would have Derek McInnes' men as favourites to repeat the two wins they have already earned in derbies this season.
However, over the course of the season, he would have his doubts about their ability to get the job done against the rest of the teams, backed up by Saturday's draw with Dundee.
More often that not, he would back Celtic to find a winning goal if they found themselves in those circumstances.
Speaking on the Go Radio Football Show, Mulgrew said: "They've done it that many times last year under Martin O'Neill. I just feel that they've got that mentality to score at the last minute, and I'm still questioning Rangers, whether they do.
"It's funny, when I think of Celtic playing Rangers tomorrow, I think Rangers are favourites in my mind. But when I think of Celtic playing any other team, and Rangers playing any other team, I always fancy Celtic to win. And Rangers, I'm not sure, convinced they can win loads of games away.
"I think if you put it down to the last ten minutes of a game, Celtic and Rangers both drawing, just say away to Dundee United or any away game, right now as it stands, I would fancy Celtic to go and get that last-minute winner. I feel that they've got that."