Celtic looked on course to take at least a point at San Siro before two late goals consigned them to a 2-0 defeat in their first match in Group H.
Although goalkeeper Fraser Forster was called into action four or five times, the Scottish champions might have nicked the goal their possession arguably deserved.
Striker Anthony Stokes had a shot blocked in the first half before his second-half free-kick from 25 yards clipped the crossbar before going over.
Georgios Samaras drove wide from distance just after the break while skipper Scott Brown hesitated enough when he was sent through by Kris Commons to allow Milan midfielder Nigel De Jong to make a crucial saving tackle.
Celtic still looked as though they would leave Italy with a point until defender Cristian Zapata's drive in the 82nd minute deflected off Parkhead left-back Emilio Izaguirre and ended up behind Forster.
The Englishman was beaten again four minutes later despite making a great save from Mario Balotelli's free-kick, as midfielder Sulley Muntari knocked in the rebound.
And Mulgrew could not hide his disappointment afterwards despite the encouraging performance.
"We're here to compete in this group, not just put in good performances and get beaten and come away happy," Mulgrew told BBC Scotland.
"It's just 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 AC 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 and we have to pick ourselves up."
Celtic next face Barcelona, who beat Ajax 4-0 in their opening encounter, as the Catalan giants look to avenge their 2-1 defeat at Parkhead in last season's competition.
And Mulgrew retains the ambition to qualify for the knockout stages for a second season running.
"It's only one game in," the former Aberdeen player said. "There's a long way to go.
"We've got to be positive. The next game comes around in a couple of weeks and we've got to be ready for that and quickly put this result out 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."
Winger Derk Boerrigter believes Celtic will have to be more clinical if they are to have a hope of progressing after describing the defeat as a missed chance.
The former Ajax player, who came on for Adam Matthews in the 75th minute, told STV: "We played pretty well but the most important thing about a game is to win and if you want to win you need to score.
"We got a lot of chances but didn't score. Yeah it was a missed opportunity.
"Milan has some good players and [we knew] if we didn't score then they might and they did.
"We were in control of the midfield with good possession so compliments to the team for that. But then again up front you don't get many chances and when you do, you need to score."
Despite the defeat Boerrigter believes the Parkhead men performed well enough against the Italian giants to recover and be a force in the group.
He added: "We have got a lot of games left so I think we can win some games to remain in the Champions League and that's what we will have to do."