After a tepid opening half-an-hour, Wigan took the lead through the impressive Callum McManaman before Kone increased the advantage five minutes before the break.
James McArthur scored a stunning third for Wigan and, after Lee Novak had pulled a goal back for Huddersfield, Kone seized on a defensive error to score a late fourth.
A much-changed Wigan side had the first chance of the game when Jordi Gomez, whose penalty saw off non-league Macclesfield Town in the last round, nodded over the bar under pressure on six minutes.
Huddersfield responded with Alan Lee heading off target, but clear-cut goalscoring opportunities were scarce until Wigan took the lead on 31 minutes.
McArthur's excellent pass inside Jack Hunt freed McManaman and the winger continued his run into the box before placing his shot beyond Alex Smithies.
McManaman turned provider for the Latics' second goal five minutes before half-time after another purposeful burst down the left-hand side into the penalty area.
His cut-back was met by Gomez, whose attempt was superbly saved by Smithies, only for the loose ball to fall to Kone to casually flick home, although the striker was offside from the initial shot.
Huddersfield made a double substitution at half-time by bringing on Novak and James Vaughan, and the hosts were much brighter at the start of the second half.
However, the wind was taken out of their sails on 55 minutes when McArthur exchanged passes with Gomez before driving a fierce shot past Smithies.
The Terriers gave themselves hope of a comeback on 62 minutes when substitute Novak powered home a header from Calum Woods' left-wing cross.
Vaughan nodded wide for Huddersfield, but the last word was had by Wigan as Kone pounced on Anthony Gerrard's mis-control before rifling past Smithies to put the seal on his side's smooth progress into the last eight.