Second-half goals from Federico Fazio and Kevin Gameiro had put the Spanish side ahead after cancelling out Marcos Tavares' opener, but Vrsic struck from 25 yards to give Maribor a fighting chance in Seville next week.
Sevilla coach Unai Emery had warned before the game that underdogs Maribor could offer a surprise to his team and the Slovenians were always competitive in this 2-2 draw.
Maribor dominated the first-half, barring a Gameiro shot from a tight angle that hit the side netting, and took the lead after 33 minutes.
Martin Milec stormed into the box and cut the ball back to the edge of the area where it found Tavares making a late run to confidently fire past the despairing arms of Javi Varas.
Mitja Viler could have doubled the lead five minutes before half-time, forcing Varas into action with a low shot that the goalkeeper fumbled but gathered at the second attempt.
Sevilla came out fighting in the second-half though and had a chance to equalise inside 30 seconds when Gameiro saw his a shot saved after being put through on goal before Ivan Rakitic blasted the rebound over the bar.
A minute later, they did equalise. Rakitic again was at the heart of it, breaking down the right and sending a low cross for Gameiro sliding in at the back post to convert.
The Spanish side looked transformed from their disappointing first-half and they took the lead in the 71st-minute, Fazio climbing highest in a penalty area scramble to head home Rakitic's corner.
But Maribor refused to give in and with eight minutes left a stunning passing move ended with substitute Vrsic thrashing a low shot past Varas from 25-yards.
There was still time for one more Sevilla chance but with a minute remaining Piotr Trochowski's 20-yard shot agonisingly hit the post, leaving the score at 2-2 and with both sides optimistic ahead of the second-leg.