Two first-half strikes within 97 seconds of each other left the game tied until two late goals from the substitutes provided the Toffees their first home win of the season.
A deflected effort from Franco Di Santo gave the visitors the lead in the 31st minute after a poor opening half an hour from both sides, but Phil Jagielka pounced only 97 seconds later to level for David Moyes' side.
The Toffees then completed their turnaround with only five minutes remaining as a rising Apostolos Vellios latched on to Tony Hibbert's sublime cross to nod home from 10 yards out. New boy Royston Drenthe then wrapped it up in stoppage time to put Everton into seventh.
The opening exchanges saw neither side able to gain a foothold in the game, with the hosts having the better start but with little effect as possession remained loose in the center of the park. Chances for both Leon Osman and Di Santo came and went, but neither goalkeeper was to be tested early on.
Everton was showing the greater endeavor but it was a tentative start that would prove to characterize the opening 20 minutes, with both Martinez and Moyes left aghast by their respective sides' inability to keep hold of the ball going forward.
With halftime approaching, a piece of inspiration was required and it was to come in the form of Argentine Di Santo, who, having received the ball off Ben Watson off a short corner, drove the ball towards goal and saw it deflected off Hibbert and into Howard's top corner to give Wigan the lead against the run of play.
The Toffees' response was incisive and only 97 seconds after conceding the game was tied.
A Leighton Baines corner was met superbly by the flying Tim Cahill, only for the Australian to see his header slam against the crossbar. However, with the ball free in the area, Jagielka was on hand to pounce and score his first goal at Goodison Park in the Premier League in three and a half years.
The game had gone from turgid to terrific in a matter of minutes and as the halftime whistle went, things were very much in the balance.
The Latics came out full of vigor in the second period and it was Victor Moses who was shining brightest, weaving in and out of Baines and linking well with Gomez. But with the Toffees looking dangerous on the break, nerves were jangling in all quarters.
Both sides were, similarly to the opening 45 minutes, proving largely wasteful and Moyes introduced Royston Drenthe and Apostolos Vellios in order to provide a little vibrancy into a game that had offered anything but in large part.
The fixture was becoming increasingly stretched, with neither side willing to settle for a point but the willingness of both teams wasn't matched by quality and as the clash moved into its final quarter of an hour, frustration reigned at Goodison.
But joy was to come as with only five minutes remaining, as Tony Hibbert picked up the ball on the right-hand side to loft a superb ball into Vellios, and the young Greek made no mistake, nodding home to give the home side the lead late on.
The three points were then wrapped up for the hosts as Drenthe, another of Moyes' substitutes, latched on to Denis Stracqualursi's through ball and coolly slid it past Ali Al-Habsi as Everton, despite having been poor for much of the game, took all three points.
There was some controversy prior to the games when Moyes named an unchanged side from the team that drew 2-2 with Aston Villa last weekend, leading Louis Saha, not even on the substitutes bench, to take to his twitter account and state somewhat mysteriously “I am not good enough” less than an hour before kickoff.