The Scottish champions' loss at the Astana Arena on Tuesday night was as unexpected as it was potentially damaging to Neil Lennon's side, who were made to pay for slackness in defence and profligacy in front of goal.
While the robust Kazakhstan side looked limited in many aspects of their play, a dramatic turnaround in front of what is sure to be a packed house is required if Celtic are to reach the lucrative group stages of the competition for the second successive season.
Mulgrew is hoping the crowd will play their part in the Hoops' rescue mission.
The Scotland international said: "The crowd will get right behind us and we will really need to go for them.
"We were disappointed to lose but we are staying positive and looking towards the second leg.
"It was a tough game but we have to believe in ourselves for the second game.
"We we were well prepared, we knew what was coming and we know what we will get again next week but we have to stand up to it better and score a few goals."
Mulgrew knows the Parkhead club's decision to allow midfielder Victor Wanyama, striker Gary Hooper and defender Kelvin Wilson, the spine of last season's team and arguably the three best players in their respective departments, to leave will come under scrutiny should they fail to overcome the Kazakh champions.
Lennon has brought in Amido Balde, Virgil van Dijk, Derk Boerrigter and Steven Mouyokolo, while Schalke 04 striker Teemu Pukki is on the radar, but the Hoops manager continues to search for more players to bolster his squad although no new signing would be eligible for the second leg in Glasgow.
"We are not using that as an excuse," said Mulgrew.
"We knew these questions would come but we have a lot of belief in each other, we have a big squad and enough players to deal with the challenge.
"You are hoping you get a more positive result but it could have been worse.
"We have to look at the positives and we have a real chance if we approach it positively next week.
"We are not thinking along the lines (of non-progression).
"We knew this tie was over two legs and we are looking to next week to turn it around."
- Sports & Recreation
- Neil Lennon