The Kazakh minnows beat the Scottish champions 2-0 at Astana Arena and Mulgrew admits the backing of a vociferous crowd at Celtic Park would be required next Wednesday if they are to reach the group stages for another season.
Celtic stunned Barcelona 2-1 at Parkhead last season and 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. We we were well prepared, we knew what was coming and we know what we'll get again next week but we have to stand up to it better and score a few goals."
Midfielder Victor Wanyama, striker Gary Hooper and defender Kelvin Wilson have been allowed to leave the club and the team which stunned the Catalans.
Manager Neil Lennon has brought in Amido Balde, Virgil van Dijk, Derk Boerrigter and Steven Mouyokolo, while Schalke 04 striker Teemu Pukki is on the radar.
"We are not using that as an excuse," said Mulgrew on the changes to the starting XI.
"We knew these questions would come but we have a lot of belief, a big squad and enough players to deal with the challenge.
"We were looking to 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."