The Hoops lost 2-1 to Aberdeen at Pittodrie to end their unbeaten run for the Scottish Premiership season as well as having their potential British record for not conceding a goal denied.
Mulgrew admits it has been tough to lose two records in the one game but insists that the Glasgow giants have to recoil swiftly from the defeat.
"What happened on Tuesday has played on our minds for a few days, but we need to put the result behind us and move towards the next match," he told Celtic TV.
"Now we just need to focus on picking up as many points as we can. At a club like Celtic you need to bounce back. The fans want to win every game and they demand you give your all.
"The manager does as well, every day in training he is on us and it shows this season, with the unbeaten run."
League leaders Celtic get the chance to atone for their aberration when they host fourth-placed Inverness on Saturday.
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