The Scottish champions lost 2-0 to little-known Shakhter Karagandy in the first leg of their Champions League playoff qualifier in Astana on Tuesday night and will need to outscore the Kazakhstanis by three goals in the return bout in Glasgow next Wednesday.
Mulgrew feels with the assistance of the Celtic Park faithful acting as a 12th man, the Hoops will be able to get the job done.
"The crowd will get right behind us and we will really need to go for them," he said.
"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 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."
He added: "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."