Hibernian lose Intertoto first leg
The Swedish side, who currently sit second in their domestic top flight and have lost only one match during their opening 12 league matches, were by far the superior side in the second-round meeting.
Goals in either half from Emir Bajrami and Andreas Augustsson put the visitors in the driving seat ahead of the return leg in Boras next Saturday.
Boss Mixu Paatelainen insisted Hibs' European adventure is not yet over after the match.
"It was visible today which team was ready to play and we all knew that. Next week we're a little bit better off for match fitness and knowledge.
"We can score over there, absolutely. We'll try our best to turn it around and we'll go there to try and win the game. If we score three goals and don't concede, we go through.
"We want to do as well as we can but it will be tough again. But we'll be better off."
Elfsborg manager Magnus Haglund was content with the result but felt his team should have scored more.
He said: "It was a good start to the two matches, 2-0 at Easter Road is a very good result for us.
"Actually, we could have scored even more in the second half but of course I'm happy with this win."