The Magpies matched their Premier League opponents throughout and after Hunt's 75th minute header, Caldwell bundled into his own net 12 minutes from time to ensure a date in round five with Fulham.
County went into the replay as the lowest ranked team left in the competition but in front of a paltry crowd they refused to lie down.
Wigan began well enough with former Swansea duo Jordi Gomez and Jason Scotland combining to create the first chance in the 11th minute.
Gomez sent struggling striker Scotland clear, but as he set himself to pull the trigger Kasper Schmeichel raced out of goal to avert the danger.
The Magpies, with director of football Sven Goran Eriksson watching on, had their first chance in the 27th minute when Ben Davies, who scored a free-kick in the 2-2 draw at Meadow Lane, saw another set-piece cleared by Caldwell.
Lee Hughes warmed Wigan keeper Vladimir Stojkovic's hands soon after as the game finally came to life.
Stojkovic then kept out another Davies shot and Ricky Ravenhill saw a shot cleared off the line by Scott Sinclair.
And Schmeichel was also in action just before the break when he brilliantly palmed away a Paul Scharner shot.
The Latics were much improved after the break but squandered a host of chances within the first 20 minutes of the restart.
Twice Scotland failed to take chances, with Gomez and Scharner also going close and they were made to pay for those misses.
After Hughes had forced a fine save from Stojkovic they went ahead when Hunt rose to head in Davies' corner on 75 minutes.
And it was two just three minutes later. Another Davies cross caused chaos in the box and after Titus Bramble missed his clearance, Caldwell helped the ball into his own net.
Substitute Luke Rodgers should have added a third in the closing stages, but the damage had already been done as County moved into round five for the first time since 1992.