The sides drew 1-1 on Matchday Two and that scoreline was replicated by half-time back in Belgium, as Lasse Nilsson cancelled out Yanis Mbombo's excellent opener.
But Nilsson and Mikkel Beckmann struck in the six minutes following half-time to ensure victory for the Swedish side.
Standard had two good chances early in proceedings, with Frederic Bulot heading against the crossbar before Igor De Camargo shot wide.
Ronnie Stam's 20-yard drive was then fielded by keeper Kevin Stuhr-Ellegaard before Elfsborg finally showed as an attacking threat.
Beckmann shot wide from near the penalty spot before Anders Svensson let fly from 25 yards, forcing Yohann Thuram-Ulien into a flying save.
But shortly after the half-hour mark, Mbombo lit up the game with a glorious curling effort into the top corner from 20 yards on his first European appearance.
Beckmann shot wide from distance but Elfsborg's equaliser arrived four minutes before the break when Viktor Claesson threaded a superb pass through the defence and Nilsson fired into the bottom corner.
And just 30 seconds after the interval, he scored his second and put his side ahead after the ball broke to him just inside the box.
Thuram-Ulien made a good save from Simon Hedlund's drive but was soon culpable for Elfsborg's third goal as Beckmann's volley, from Nilsson's cross, squirmed through his grasp.
Svensson and Nilsson both fired over from long range but Standard could have pulled a goal back in the 73rd minute when substitute Dudu Biton hit the woodwork when well placed.
Stuhr-Ellegard saved another Biton effort, from Stam's corner, and Standard could find no way back, Elfsborg going closest late on when Marcus Rohden's drive was parried by Thuram-Ulien.