Though Sweden were keen to earn a first win in the group and first under Potter, the playoffs were on the English manager's mind, as he started only one of the nine players currently sitting one yellow card away from a suspension - Emil Krafth.
That meant a place on the bench for Alexander Isak, and the early chance for Sweden instead fell to Gustav Lundgren, whose powerful drive flew narrowly over the crossbar eight minutes in.
At the other end, Slovenia had their first opportunity five minutes later, coming in the form of a venemous Zan Karnicnik strike, which the deputising Jacob Widell Zetterstrom managed to palm over his crossbar.
Just before the half-hour mark, space opened up through the middle for Mattias Svanberg to have a go from range, but he too failed to keep his effort down.
The first-half was proving low on quality but it was an end-to-end contest in Solna, as soon after Krafth justified his selection with two key blocks in a matter of seconds, before Erik Janza latched onto the loose ball and became the latest player to send the ball sailing over.
That looked to be the final decent opportunity of the half, until added time, when Adam Gnezda Cerin let fly from over 25 yards, but Widell Zetterstrom managed to get his fingertips to it and preserve the first-half parity.
Blagult made a bright start to the second half but when Roony Bardghji failed to find a route to goal, Besfort Zeneli could only send his follow-up off target. From then on, the visitors began to take control, though Widell Zetterstrom had no issue with Timi Elsnik's effort with the hour-mark approaching.
However, the Swedish goalkeeper had no chance with Elsnik's next chance in the 64th minute. Under little pressure, Carl Starfelt passed the ball straight to Andraz Sporar, who got to the byline and cut the ball back for the Crvena Zvezda man to tap Slovenia in front.
Having conceded the opening goal for the fifth match in succession, Sweden finally tested Igor Vekic in the Zmajecki goal eight minutes after falling behind - Daniel Svensson's header from out wide was met with a firm Isac Lidberg header, and Vekic had to be at full stretch to keep it out.
The home side were running out of time to force an equaliser, but three minutes from time, a deep Alexander Bernhardsson cross was kept alive at the back post by Gustaf Lagerbielke, whose knockdown was volleyed home by Lundgren.
Slovenia had one final chance to win it in added time, but Widell Zetterstrom reacted well to deny Tomi Horvat to secure a share of the spoils. Matjaz Kek's men finish on four points in the group, having won zero and drawn four of their matches, two more than Sweden managed over six games each.