Marc-Antoine Fortune failed to make the 16-man squad for the trip to Fife but the visitors still finished the game with four strikers, Rasmussen, Georgios Samaras, and Diomansy Kamara joined after the interval by Robbie Keane with the latter two new boys scoring their first goals for the club.
Ahead of Wednesday night's SPL clash with Hearts at Celtic Park, Rasmussen, who cheekily tried to claim Callum Woods' own goal, said:
"It is important to score goals when you are new in the team," he said. "I am definitely claiming two.
"I was going after the ball and I think I got a little touch as did the defender so I will take the goal and that makes it two!
"But it was important that we won and got through to the next round.
"Celtic are a big club, there should be a lot of good players and that is not a problem for me.
"It is not like you can put 11 strikers on the pitch and score 11 goals but it is good to have a lot of good players.
"We just have to fight in training to get a place in the team."
After Kamara opened the scoring for the visitors, David Graham's strike and Andy Kirk's penalty put the home side ahead before Rasmussen levelled two minutes from the interval.
However, any thoughts of a possible shock ended in the second half after Woods turned the ball into his own net under pressure from Rasmussen before Keane, on for Marc Crosas, scored his first goal for the club from the spot after he had been felled by Pars' skipper Austin McCann.
Dunfermline manager Jim McIntyre was happy enough with the effort of his players.
He said: "I said to the players before the game that they would need to believe that we could get a result and if they didn't then they could end up on the end of a hiding.
"So I was very pleased by the way we approached the game and I was really pleased by the way we came back after losing the goal."