Matt Target's early opener put Villa in front, but Leicester enjoyed the majority of possession and chances and finally found a way past Orjan Nyland when Kelechi Iheanacho levelled 18 minutes from time.
However, Villa snatched victory and a trip to Wembley on March 1 to face Manchester City or Manchester United when Trezeguet connected perfectly with fellow Egyptian Ahmed Elmohamady's cross three minutes into added time.
Defeat ends Leicester boss Brendan Rodgers's run of winning 31 straight domestic cup ties after a dominant spell at Celtic and he will rue his side's inability to turn their dominance of the game into goals.
The visitors started with top scorer Jamie Vardy on the bench as he was eased back into action after a glute injury, but the Foxes still blasted out of the traps.
Iheanacho was denied in the first of a string of big saves by Nyland in the Villa goal inside the first three minutes.
James Maddison then had a hat-trick of efforts, two of which brought saves from Nyland.
However, it was Villa who struck first when the impressive Jack Grealish teed up Targett to fire across Kasper Schmeichel 12 minutes in.
Nyland may not even have played if Villa's new number one Pepe Reina had been fit to play, but the Norwegian was inspired as he then produced his best save of the night to turn Youri Tielemans's piledriver onto the bar.
Maddison was furious moments later when he saw another effort repelled by the arm of Marvelous Nakamba.
Mbwana Samatta was making his Villa debut after a £8.5 million ($11 million) switch from Genk, but his first experience of Villa Park is not one the Tanzanian will remember fondly as he somehow failed to turn home another brilliant Grealish delivery early in the second half.
That could have proved the knockout blow for Leicester, but instead they were very much back in the tie when an explosive burst from Harvey Barnes teed up Iheanacho to finish.
Jonny Evans and Maddison then came close to putting Rodgers's men in front.
But just when both sides seemed to be preparing for extra-time, Leicester were hit with a sucker punch when Elmohamady's cross was volleyed into the bottom corner by his compatriot.
Trezeguet had to be hauled from the field by Villa security personnel after he was mobbed in a pitch invasion after the final whistle as the home fans celebrated wildly.