But the home side were only in a position to strike late on after a costly mistake from Sweden goalkeeper Robin Olsen early in the second half, as he let a tame Sandi Lovric shot slip through his fingers for the first of two Slovenian equalisers in the match.
"The disappointment is huge, as big as it can be," the goalkeeper said in the mix zone after the match.
"This is something that must not happen. It's the worst a goalkeeper can do. It's clear that it's tough."
A veteran of 78 caps, Olsen made it clear there was no one to blame but himself for the goal that cancelled out Anthony Elanga's opener in the first half: "There are no excuses, whether it bounces or wobbles. It's a ball I should take."
The Malmo goalkeeper admitted it will be difficult to move on from the mistake that means Sweden are already two points behind Switzerland after one match each in Group B, but his teammates are backing him to put things right ahead of Monday's meeting with Kosovo.
"It will be up to the room, hiding and grieving. Then I have to wake up, hopefully with new energy. Of course, it will be a disappointment - it should be. It's just as tough every time."
"There's no other option (other than facing Kosovo). I've received incredible support from everyone in the team."
Sweden looked for a while that they would leave Slovenia with all three points after Yasin Ayari restored their advantage with 17 minutes left to play.
"I think that's approved," the 21-year-old said of his performance. "In the game maybe I should be a bit more involved, it's a bit crowded in the middle, but it's okay."
The Brighton & Hove Albion midfielder is proving to be an important attacking player for manager Jon Dahl Tomasson, who kept Alexander Isak on the bench, while Viktor Gyokeres had an off night in front of goal.
"Since he has only had three trainings and no preseason, and of course no match time, the risk was too great to use him today," Tomasson said to the media of Isak.
Review Slovenia 2-2 Sweden and all World Cup Qualifiers on Flashscore.