The La Liga outfit appeared on course to establish a potentially-decisive advantage, but Vardy scored 17 minutes from time to keep the Premier League champions in the tie.
Leicester struggled for long periods of the match in Spain, but manager Claudio Ranieri will have been delighted with the response of his side after falling two goals behind.
Up the other end, Christian Fuchs fired well over from distance and for the remainder of the first half, it was all Sevilla and they should have taken the lead before the quarter-hour mark.
Schmeichel also had to be alert to save from Sergio Escudero but he was powerless to prevent Sevilla from taking the lead when Sarabia powered a header into the bottom corner after meeting Escudero's cross.
A minute later, a deflected effort from Stevan Jovetic almost caught out Schmeichel from 40 yards, before the Danish stopper saved well from Correa who had tried his luck from the edge of the penalty area.
In added-on time in the first half, Vitolo headed over from close range after getting in the way of Correa, who was perhaps better placed to add a second for his side.
After the restart, Leicester started to show some ambition and Wilfred Ndidi had a shot from 20 yards saved by Sergio Rico, but they had a lucky escape when Vitolo's shot from an acute angle struck the post and went across the face of goal.
However, shortly after the hour mark, Sevilla doubled their lead through Correa, who fired into the roof of the net from 12 yards out after good work from Jovetic.
It appeared that Leicester would do well to return to the King Power Stadium facing a 2-0 deficit but against the run of play, they netted an away goal which had seemed so unlikely throughout the opening 72 minutes.
Danny Drinkwater was played down the left-hand side of the penalty area and his pin-point cross picked out Vardy, who converted into the roof of the net from six yards out.
Leicester still had ambitions for a second but it soon became apparent that they would be best served trying to hold on to what they had as they dug in to try to take something back to the East Midlands.
Jovetic saw a shot deflected wide of the post with Schmeichel beaten, while Adil Rami powered a header off the crossbar with just two minutes left on the clock.
Sevilla mounted one last push in added-on time but Leicester were able to see out the final three minutes to give themselves a realistic chance in the second leg.