Rasmussen made the crucial intervention by netting a late winner but he claims the Parkhead faithful were just as important.
Celtic were heading for more disappointment in a difficult season with just 10 minutes left when Hibernian led 2-1.
But Marc-Antoine Fortune and substitute Rasmussen sealed a turnaround as the fans responded to their team's sustained pressure and the Dane felt that, despite the swathes of empty seats, they helped drive them to three points.
"I think the crowd were unbelievable in the last 20 minutes and they really helped us to get a win," he told the Celtic View.
"Every time we play we have a very good crowd behind us.
"But when they had the possibility to give us something extra they did, and it helped us very much so that was nice."
While the fans were roused as the match developed, caretaker boss Neil Lennon was animated throughout and sprinted 30 yards to celebrate Rasmussen's 87th-minute winner with the players.
The Denmark international believes such a display of bonding is symptomatic of the spirit engendered by the former Celtic skipper.
"He's been great setting the players up for training and matches," said Rasmussen, who is suspended for Sunday's Clydesdale Bank Premier League game against Dundee United at Tannadice.
"I really like him. I think we as a team, and me personally, can learn a lot from him.
"I also think one of the things that Lenny has tried to create is a good team spirit and attitude, and that's getting better and better."
Northern Ireland winger Niall McGinn has also been boosted by the impact of Lennon after starting the last two matches following more than two-and-a-half months out of the team amid injury problems.
"I am more than happy to be back and to go straight into the team has been unbelievable," McGinn told The Huddle Online.
"A lot of thanks has to go to Neil Lennon for putting me straight in the team again.
"I enjoyed my time working with Neil at reserve level and it's been great getting to work with him in the first team as well.
"He's been behind me 100%, ever since I joined the club."