It was a tough evening for Neil Lennon's side against the well-organised visitors but Kris Commons' second-half header gave the Scottish champions a narrow advantage to take to Boras for the second leg.
Elfsborg's English forward, James Keene, insisted that the Swedish champions have a "great chance" of overturning the deficit.
The former Portsmouth player said: "We are going into the game at home with confidence.
"Celtic Park was always going to be a tough place but I think we showed that we are going to ask a lot of questions at home.
"We are pretty happy with 1-0. We played pretty well. It is going to be a different game when they come to our place.
"I think we have a great chance, it is definitely not over."
Elfsborg boss Jorgen Lennartsson also believes his side are still in the tie.
He said: "They were the better side and deserved to win the game. But I don't think Celtic should feel that the tie is over because it is not."