Trailing 2-1 from the first leg, after which Claudio Ranieri was sacked as manager, Leicester scored through Wes Morgan and Marc Albrighton before Schmeichel repelled Steven N'Zonzi's spot-kick.
It gave them a 3-2 aggregate win and continued a remarkable turnaround under new manager Craig Shakespeare, who has won three games in a row since stepping up from his role as Ranieri's assistant.
"The players can be immensely proud of themselves," said Shakespeare.
"When we needed Kasper the most, he came up trumps. We deserve to be there. We have just knocked out one of the best teams in Europe in my opinion."
Unrecognisable in the defence of their miraculous Premier League title, Leicester have found solace in the Champions League and have now won all four of their home matches in the competition.
