The Scottish champions were able to keep a clean sheet against the Swedes, who have scored 19 goals in their past six outings in all competitions, and Hoops manager Neil Lennon was thrilled with his sides performance.
"I am delighted with the performance and the result," Lennon said. "They made it difficult for us to break them down but it was not frustrating, it was understandable.
"We weren't at our best but that was always going to be the case, there is still a bit of rust there. We still have a lot to do in the tie but we are confident.
"The crucial thing is the clean sheet. That is important. If we score over there Elfsborg have got to have three. The tie is probably in the balance but we are confident that we can go there and get a result.
"I would have taken 1-0 before the game."