The Terriers secured the match thanks to Sean Scannell's second-half winner, but all the attention was on the club's vacant managerial position after they confirmed they will begin talks with Coventry manager Mark Robins to become Simon Grayson's replacement.
They went ahead in the fifth minute on Tuesday night when Adam Clayton curled in a fine free-kick, but they led for just two minutes before Michael Keane levelled from a set-piece.
Nigel Pearson's Leicester were in charge but they were hit on the break by Scannell's fine solo goal with 15 minutes remaining, and despite a late onslaught Huddersfield were able to hold on.
They booked a home clash with Wigan on Sunday with sights on a Premier League scalp, while Leicester are left to concentrate on their Championship promotion bid.
Wes Morgan brought down Scannell and Clayton stepped up and fired a 25-yard free-kick past Kasper Schmeichel.
But their joy was short-lived as Leicester hit back just four minutes later.
Ritchie De Laet got on the end of Paul Gallagher's corner, and after his header was blocked, Keane was on hand to poke home his first senior goal.
Leicester regained the momentum after the restart and Richie Wellens flew a shot wide while Marko Futacs had a close-range effort blocked following a dangerous corner.
However, they were unable to make their domination count and just as news of Huddersfield's contact with Robins was breaking, the visitors regained the lead in the 75th minute.
Scannell skinned Matty James on the left following a counter attack and he surged into the penalty area before slotting home.
Pearson's side threw everything forward in an attempt to force extra-time, with Jamie Vardy heading the best chance wide as Huddersfield advanced to Sunday's home tie against Roberto Martinez's Latics.