The away side had not beaten Everton in their last nine attempts, but they went on to rip up the form book with three unanswered goals during four first-half minutes.
Everton went in search of goals after the restart, but they were unable to find a way through the Wigan defence.
You can find out how the match unfolded with our minute-by-minute commentary below.
"It's a good situation to have to break up the league fixtures with what could be a very significant moment in the history of the club," the Austrian told the club's official website.
"If we can play a good game at Goodison Park and get a victory there then we will be able to pick up our confidence again and bring it back to the league games."
"I’m aware of everything that happens at Wigan and I want to improve it all," Martínez is quoted as saying by The Telegraph. "I like the fact my life is this football club, it is a project I’m very proud of.
"This is the first time we have been successful in cup competitions and that is down to the readiness of the young players. We put a lot of emphasis on the 18 to 21-year-olds trying to develop them. Previously we couldn’t compete when we made two or three changes.
"It has made the squad bigger and stronger. I can rotate. It shows the work we’re doing behind the scenes is paying off. Our main prize is surviving in the Premier League, but we can enjoy this cup game because the youngsters coming through.
"We made a profit this year for the first time and we will be moving into a new training ground, which is massive for the club. That enables us to keep growing. A certain percentage of the first team squad should be made up of players from the area, with foreigners who bring something different."
EVERTON: Springthorpe, Gibson, Naismith, Hitzlsperger, Barkley, Anichebe, Duffy
WIGAN: Al Habsi, Stam, Golobart, Di Santo, Henriquez, Miyaichi, McArthur
"Everton have both scored and conceded in 13 of their last 17 FA Cup games."
"Everton striker Nikica Jelavic has netted just one goal in his last 19 hours and 38 minutes of football for the club."
"We have always had tight games against Wigan and I'm sure it will be the same today. Their is a big prize for the winner. Wigan have been playing well and they don't always get the results. There were signs last week that we were getting our form back."
"Paul Scharner has scored five goals against Everton - he has no more than two against any other side."
For a defender, that is some record. The Austrian is currently having a proper old school battle with Marouane Fellaini. They don't look to be the best of friends right now, put it that way!
"That was Wigan's best 45 minutes of the season. For all Everton's good football this season, they have produced their worst half of football.
"Everton will have to make some changes at half-time, or else Moyes just giving them the proverbial rocket. I will be astonished if they can turn this around from here, but of they do this game will be a cracker."
"They say its a game of two halves. Well time to show what the Blues are made of."