With the first leg having finished goalless in Prague, Leicester were strong favourites to reach the last 16 at the King Power Stadium.
But Lukas Provod gave Slavia the lead early in the second half with a cool volleyed finish inside the penalty area after his run went unchecked by the Leicester defence.
Abdallah Dipo Sima made the outcome certain with 11 minutes left when he fired home spectacularly from just outside the box as the Premier League high-flyers went crashing out.
The Foxes tried to seize the early initiative as Cengiz Under twice struck efforts from distance, but their opponents looked the more threatening side in the first half.
A weak Caglar Soyuncu header from a Slavia long ball allowed striker Sima to break clear just before the half hour but his early shot went wide of Kasper Schmeichel's goal.
The unmarked Jan Kuchta should have done better for the Czech First League leaders when he planted a header wide from a good position.
Moments before the interval Jamie Vardy escaped down the left but with few options in the box he kicked the advertising hoardings in frustration after his cross was easily gathered by Ondrej Kolar.
Slavia took a deserved lead four minutes into the second half when Nicolae Stanciu's cross picked out the run of Provod, who side-footed a volley beyond Schmeichel.
Rodgers missed the creative talent of the injured James Maddison but he did bring top scorer Harvey Barnes off the bench just after the hour.
However, Slavia grabbed a second when the impressive Sima was allowed time and space by the Leicester defence and curled a powerful low shot into the corner of the net.