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.
Hibs, back in pre-season training for just a fortnight and playing their first competitive game, were sadly lacking in match sharpness and devoid of inspiration as they surrendered a sizeable advantage.
Elfsborg, well-organised in defence and physical and threatening in attack, now look strong favourites to progress to a meeting with Latvian outfit FK Riga or Dublin's Bohemians in the battle for a UEFA Cup spot.
Given Hibs' short summer break of barely a month, it was hardly surprising the visitors made the stronger start.
Just three minutes had elapsed when Denni Avdic fed the ball through to Daniel Mobaeck and the striker was only denied when Yves Ma-Kalambay dived bravely at his feet.
Hibs began to settle and after nine minutes Ian Murray drove forward on the left, feeding Dean Shiels near the by-line.
The Northern Irishman did brilliantly to twist and swing a cross towards the near post where Scotland star Steven Fletcher flashed a header over.
However, the Swedes grabbed a well-worked away goal as early as the 16th minute.
Former Southampton midfielder Anders Svensson, who started all off Sweden's Euro 2008 matches, flicked a neat ball down the left which eluded the Hibs rearguard.
Bajrami ghosted into the penalty area and calmly opened his body to place a neat shot across the helpless Ma-Kalambay and into the right-hand corner.
Hibs sought a quick response and Shiels was again involved, this time on the left, as he cut the ball back for the on-rushing Filipe Morais.
The Portuguese midfielder took aim from 18 yards but Johan Wiland, Sweden's back-up keeper at the summer finals, dived to his right to make a superb stop.
Svensson, by some distance the best player on the park, then clipped a sublime ball over the Hibs defence but Mobaeck sliced wide with only Ma-Kalambay to beat.
The lively Shiels tried his luck in the 39th minute, cutting in from the right to drive a left-footed shot towards Wiland that he plucked out of the air.
Mixu Paatelainen made a change at the break, taking captain Rob Jones off and replacing him with Martin Canning at centre-half.
The new arrival was soon involved, stopping a near certain goal when he deflected Avdic's near-post effort wide.
Hibs' lack of sharpness remained evident and Bajrami was next to be denied, this time Ma-Kalambay rushing out to the edge of his area after the goalscorer had made a surging run.
The home support were growing frustrated at Hibs' failure to gain at foothold and Brian Kerr was jeered off as Ross Chisholm came on after 55 minutes.
The Swedes remained in the ascendancy and centre-half Augustsson came close to a decisive second goal when he rose unchallenged to power in a 10-yard header that John Rankin nodded off the line at the near post.
Fletcher offered hope on 63 minutes, latching on to a loose ball to turn and strike a left-footed 20-yard effort that seemed destined for the bottom corner until Wiland pushed out a hand.
It was to prove a key save as Elfsborg immediately raced back up the pitch to double their advantage with a goal straight from the training ground.
From a corner, Svensson rolled the ball towards the edge of the penalty area and Augustsson took advantage of poor marking to fire home a low drive from 20 yards that appeared to take a deflection as it eluded Ma-Kalambay.
Mobaeck soon passed up a golden opportunity for a third as he screwed wide from the corner of the six-yard box.
In injury-time, Avdic cannoned a shot off the post to spare Hibs more blushes.