The reigning Scottish champions can thank Kris Commons who spared their blushes with an injury-time leveller against the Highland outfit who took the lead early in the second half courtesy of Richard Brittain's superb free-kick.
Although it was a close call against a much less fancied opponent, Mulgrew was still proud of the Celtic team who stuck together in order to fight it out until the death for a result.
"Maybe a year ago we might have lost that game," he said.
"But we stayed strong and fought to the end and got the point and we were happy with that in the end.
"When you are on the pitch you always believe you will get something and if we had scored earlier we might have went on and won it but we took a point in the end.
"I think everyone believed and that's why we got the goal."