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Having survived a series of Wigan attempts on goal, League Two promotion hopefuls County took a shock lead with 15 minutes to go as Stephen Hunt headed home Ben Davies's corner.
Gary Caldwell then scored an own goal three minutes later to leave the Premier League side stunned and the Magpies looking forward to a trip to Fulham a week on Sunday.
It was an inspirational victory for cash-strapped County, the lowest-ranked team left in the competition, and well deserved.
The visitors had chances from the early stages of the match. They won a free-kick in the seventh minute when Titus Bramble bundled Lee Hughes over, but Johnnie Jackson's delivery was easily cleared by Caldwell.
Seconds later Jordi Gomez found some space midway into the County half and tried a speculative effort from 30 yards which Kasper Schmeichel had little trouble dealing with.
Gomez then sent Emmerson Boyce through on the right but the defender shot wide, and Jason Scotland, playing as a lone striker, looked as unconvincing as he had done on Saturday against Everton after getting himself into a one-on-one situation with Schmeichel, hitting a tame effort straight at the goalkeeper.
The Magpies began to assert themselves just before the half hour and Mike Edwards sent a looping header over the bar from Davies's corner before Hughes stung Vladimir Stojkovic's palms with a fierce shot taken just outside the box.
Scott Sinclair responded at the other end, dribbling past several defenders and sending a low drive across Schmeichel's goal, but it was County on the attack again a minute later as Davies brought another save out of Stojkovic.
The Serbian goalkeeper failed to deal with the resulting corner and the ball fell to Neal Bishop, who was unlucky to see his shot deflected wide.
Paul Scharner then hit a volley at Schmeichel, but once again the effort lacked conviction.
Wigan came out for the second half with renewed purpose and almost broke the deadlock less than a minute after the restart.
Scotland was again unable to convert from a good position, although this time it took an impressive save to deny him, Schmeichel clawing the striker's close-range header away for a corner.
Five minutes later Gomez swung in a useful cross for Scharner at the far post, but it was just behind the Austrian, who was unable to direct his header properly.
Sinclair then wasted a glorious opportunity chasing a through-ball and colliding with Schmeichel.
The winger got up quickly and had an empty net to shoot at while Schmeichel remained on the ground, but his snap-shot was off-target and County survived.
Scotland then saw his drive deflected wide and Gomez struck a bobbling effort which Schmeichel turned behind for a corner.
With his team seemingly in the ascendancy, Roberto Martinez brought on some fresh legs in the form of Charles N'Zogbia and James McCarthy, but it did not improve the standard of the finishing.
Scotland sent another effort wide after being fed by Boyce and Scharner had the ball in the net, but only via a hole in the side-netting.
Sinclair and Gomez were next to fire in at Schmeichel, but on both occasions there was insufficient power and the goalkeeper saved comfortably.
It seemed only a matter of time before Wigan scored, but with 20 minutes remaining Hughes curled a wonderful effort out of nowhere which Stojkovic had to fully extend himself to push away.
Gomez tried to hit back with a drive that went high and wide but County almost made it 3-0 shortly afterwards as substitute Luke Rodgers flashed a shot narrowly over.