The Hoops more than held their own against the seven-time winners until two late goals consigned them to a 2-0 defeat in the Group H opener.
Celtic enjoyed large spells of possession and came close to opening the scoring through Kris Commons and Georgios Samaras before Anthony Stokes rattled the bar with a 25-yard free-kick nine minutes from time.
However, just 60 seconds later Emilio Izaguirre deflected a long-range shot beyond the reach of Fraser Forster and an 86th-minute Sulley Muntari goal put the tie beyond the visitors.
But despite the manner of the defeat, Mulgrew is confident Celtic can pick up enough points in the remaining five games to at least match the heroics of last year when they advanced from the group stages along with Barcelona.
"We're here to compete in this group, not just put in good performances and get beaten and come away happy," he told BBC Scotland.
"It's a massive disappointment. I thought we played well and maybe just couldn't take our chances. We never deserved a 2-0, but we know that's what Milan can do.
"It felt quite comfortable and maybe if we had taken one of our chances it would have been a different result.
"But that's the Champions League for you. It's only one game, there's a long way to go and we'll pick ourselves up.
"We've got to be positive. The next game comes around in a couple of weeks and we've got to be ready for it and quickly put this out of our system.
"I think we're growing well in the competition. There's a lot of belief in the squad - the manager installs that in us - maybe just a bit of belief missing in front of goal."
Celtic have two home games against Barcelona on October 1 and Ajax (October 22) before travelling to Amsterdam on November 6.