In what was the Swedish hosts' first game since Klas Ingesson had been promoted from youth coach to take temporary charge of them following Jorgen Lennartsson's dismissal on Monday, they went ahead in the 23rd minute through Viktor Claesson's impressive curling effort.
Standard were level just after the hour mark thanks to a left-footed finish by Geoffrey Mujangi Bia.
The Belgians' last Europa League outing saw them open their Group C campaign with a shock 2-1 home loss to Esbjerg, who were beaten 2-1 on Thursday at home by Salzburg, the 4-0 victors over Elfsborg in the first round of matches.
Elfsborg made a lively start against Standard, with Eiji Kawashima parrying attempts from range by Lasse Nilsson and Claesson.
Anders Svensson tried his luck three times to no avail, while at the other end, Mujangi Bia sent a header into the arms of Kevin Stuhr-Ellegaard and Imoh Ezekiel's shot was blocked by Sebastian Holmen.
Claesson, 20 yards out, then fired in from Stefan Ishizaki's corner.
Two efforts from Mujangi Bia towards the end of the first half did little to trouble Stuhr-Ellegaard, but the midfielder then had the ball in the net in the 62nd minute as he made the most of a clever pass from Michy Batshuayi.
In the 68th Mujangi Bia shot wide from inside the box, and Elfsborg's later attempts to get back in front were fruitless as Johan Larsson drilled a free-kick wide and substitute Mohamed Bangura's strike was comfortably dealt with by Kawashima.