The Hoops' stand-in skipper was speaking on the day that Legia Warsaw 's appeal against UEFA's decision to exclude them from the competition for fielding a suspended player had been rejected.
The Polish club were found to have fielded an ineligible player, Bartosz Bereszynski, in the second leg of their third qualifying round clash against the Scottish champions which they won 2-0 to follow a 4-1 first-leg success.
Ronny Deila's side were duly awarded a default 3-0 win and progressed on away goals and will meet Maribor in the first-leg of their play-off in Slovenia next week.
Mulgrew insists the Scottish champions have to concentrate on taking advantage of their good fortune, rather than feeling too sheepish at the way in which they were reinstated.
"We are grateful to get another chance," he said.
"It was disappointing how the two games (against Legia) went but rules are rules and we are happy to be back involved.
"Maybe there is a wee bit (of embarrassment) but what can you do?
"You have to quickly get over that and get on with it. You can't feel sorry for yourselves and be embarrassed at getting through.
"It is not the way we wanted to get through but we have to get on with it.
"We will definitely be up for it and determined to put up a much better show."
Mulgrew acknowledged the rarity of being involved in his second such UEFA controversy in four years.
He was in the Celtic side which benefited from a blunder by Sion four years ago.
The Swiss club had qualified from the play-off stages of the Europa League after defeating Celtic 3-1 on aggregate but were ruled to have used ineligible players and consequently thrown out, with the Hoops progressing to the group stages.
"I was saying that to the boys, it is unreal," said the Scotland international.
"You would be lucky to happen once, to happen twice is unbelievable but you have to get on with it."
Mulgrew, who will be back in the side for the visit of Dundee United on Saturday following a one-game suspension, did express sympathy for his fellow professionals at Legia.
"I understand their disappointment," he said. "We would have been absolutely devastated had the shoe been on the other foot.
"We are in a bit of an awkward position, there is nothing we can do about it. It is not our fault."
The Polish side's appeal was heard by UEFA's Appeals Body and after receiving the decision, club owner Dariusz Mioduski confirmed the club would take the matter to the Court of Arbitration for Sport.
Legia believed Bereszynski's suspension had been served after he missed both legs of the previous qualifier against St Patrick's Athletic as well as the first-leg against Celtic, but their failure to register him for the St Pat's tie proved costly.
Mioduski released a statement through the club's official website which stressed Legia's determination to keep fighting for Champions League reinstatement.
The statement read: "Till the last moment we hoped that the spirit of sport wins, we find the decision extremely unfair.
"However we expected that decision and were prepared for it. We are to appeal to the CAS promptly.
"We will use the full appeal procedure, in order to get back our result that was fairly gained on the pitch.
"We will certainly fully subordinate to the final verdict. But I still hope that football will win, as well as the values it is based on: fair-play and respect to the on-the-pitch result."
Before the verdict, Hoops manager Ronny Deila spoke only about facing Maribor in the play-off where he hopes his side take advantage of playing at Celtic Park, after taking the previous two European home games to Murrayfield in Edinburgh while their own stadium was out of commission due to its use in the 2014 Commonwealth Games.
"We have prepared for Maribor," said the Norwegian. "They are a good team and they have been the champions of Slovenia in the last three years.
"It is hard to say if they are better or worse (than Legia Warsaw) but they are a good team.
"Legia was a good team as well, so nothing is easy but we can do much better than we did against Legia and we will play at Celtic Park and those two things can be good for us.
"I have seen European nights at Celtic Park and I understand why you get results there and that's because you have unbelievable fans behind you.''