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The former Parkhead player suggests that while Rangers might be favourites in a direct match-up, Celtic's determination often helps them clinch victories in challenging scenarios.
Mulgrew's faith in Celtic's winning mentality stems from their consistent performance in tight situations, a trait he believes sets them apart from their rivals.
Speaking on the Go Radio Football Show, he 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."
Despite his belief in Celtic's resilience, he acknowledged Rangers' strength, saying: "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."
His belief in Celtic's ability to secure last-minute victories is unwavering.
He said: "I think if you put it down to the last ten minutes of a game, Celtic and Rangers both drawing... I would fancy Celtic to go and get that last-minute winner.
"I feel that they've got that."
Mulgrew's insight reflects his confidence in Celtic's ability to consistently perform under pressure, giving them the edge in the long run.