Captain Hans Henrik Andreasen scored both goals for the visitors with tidy headers, and, although Jon Jonsson's brilliant free-kick halved the deficit midway through the second half, the hosts could not force an equaliser.
The result kept Esbjerg second in the pool on six points from three games - three points behind leaders Salzburg - and left Elfsborg with just the one point from three games.
Andreasen gave Esbjerg the perfect start, nodding Jonas Knudsen's left-wing cross into the bottom-right corner from the first noteworthy attack.
Niels Frederiksen's side continued to look the more likely to add to the scoring as the half wore on, although Peter Ankersen's shot and an Andreasen header were comfortably dealt with by Elfsborg goalkeeper Kevin Stuhr-Ellegaard.
Elfsborg mustered two decent opportunities in the final seven minutes of the first half, though, with Esbjerg goalkeeper Frederik Ronnow - who until then had been a virtual spectator - alert enough to pull off fine saves on both occasions.
He first parried Johan Larsson's fierce 20-yard free-kick and then tipped Viktor Claesson's header around the post from Stefan Ishizaki's corner to ensure Esbjerg headed in ahead at the interval.
Elfsborg continued to press forward in search of an equaliser after the restart but failed to unduly trouble Ronnow, who punched an Ishizaki drive away and then looked on as Lasse Nilsson blasted wide and Mohamed Bangura just failed to direct his header on target from Ishizaki's corner.
When the second goal arrived, however, it again fell to Andreasen, the midfielder deadly when meeting Ankersen's cross from the right in the 66th minute.
The two-goal lead did not last for long, Jonsson sending a wonderful free-kick into the top-right corner moments later after Bangura had been fouled by Kian Hansen, but that was as good as it got for Elfsborg.
Ronnow fisted away a powerful header from Bangura and got away with fumbling a Larsson attempt late on as Esbjerg held on.