Wednesday dominated the first 90 minutes taking a 32nd minute before Micky Adams side hit back with an injury-time equaliser and an extra-time winner.
The Hillsborough side turned the screw from the start, midfielder Craig Armstrong almost catching out Ian Walker in the ninth minute with an inswinging corner, the former Spurs keeper tipping it over the bar.
Minutes later Lloyd Owusu almost found a way through, Callum Davidson charging in to block his shot and deflect the ball for a corner.
Alan Quinn tried his luck with a shot from 20 yards but it packed no punch as Walker smothered the ball on the line.
Wednesday, forcing the pace, nearly broke the deadlock on 17 minutes. Leon Knight whipped in a low cross and Shefki Kuqi just failed to connect as he slid in at the far post.
Knight miscued minutes later after Kuqi had headed the ball into his path before Knight almost broke the deadlock. Kuqi picked out the little striker and Knight produced a blistering shot from the edge of the area, pushed away one-handed by Walker as he flung himself to his left.
Wednesday finally snatched the lead in the 32nd minute, the hardworking Kuqi feeding Gerald Sibon who drilled his low shot from 12 yards into the far corner.
Owusu saw his right-foot strike deflected after weaving into the area before Sibon produced another eye-catcher, a stinging shot from 30 yards which knocked Walker off his feet as he turned the ball over the bar.
Sibon had a stinging shot blocked minutes after the restart before Leicester almost levelled matters in the 53rd minute, Matt Elliott firing against the post with the Wednesday back four looking for an offside flag.
Knight tore Leicester to shreds in the 76th minute weaving his way to the edge of the area before finishing with a right-foot strike, which beat Walker but flashed a fraction wide of the far post.
Turkish international Muzzy Izzet snatched the equaliser in second-half injury time his free-kick beating the Wednesday wall and evading Kevin Pressman.
Trevor Benjamin grabbed the winner seven minutes into extra time sliding in at the near post to connect with an Alan Rogers cross.