The hosts went ahead in the 57th minute through Hugo Rodallega, who neatly guided the ball over David Stockdale.
Substitute Johnson drew things level with four minutes of normal time remaining with a shot that hit Gary Caldwell on its way towards goal, leaving Ali Al Habsi unable to stop it going in.
The result saw Wigan move out of the Premier League drop zone to 17th but they are only a point clear of the bottom three, while Fulham are one point better off in 15th.
In a scrappy opening to the game, Chris Baird made the first attempt on goal after eight minutes, striking a low shot from outside the area which Al Habsi got down to his left to save comfortably.
Two minutes later, Wigan launched an attack which saw Mohamed Diame cross for Rodallega, but the Colombian sent his effort wide.
Baird tried his luck from even further out on the quarter-hour mark after receiving the ball from his skipper Danny Murphy, but the strike went high and wide.
Damien Duff then drove off-target before Clint Dempsey broke into the Latics box, only to find himself crowded out.
In the 27th minute Rodallega went the closest yet to breaking the deadlock when he headed Ronnie Stam's cross against the post. The rebound fell to Charles N'Zogbia, whose shot cannoned off a Fulham shirt.
Soon after Stockdale flapped at a high delivery into the box with Rodallega lurking, but the ball was cleared away.
Duff then found himself through one-on-one with Al Habsi, but he could only place his attempt against the legs of the Oman stopper and Simon Davies' follow-up cleared the crossbar by some distance.
Duff was next to shoot too high and Diame scuffed the ball into Stockdale's arms.
Dempsey unleashed an effort shortly after the restart which went the wrong side of the post, but it was Wigan who started to show more urgency in their play.
Stam lofted one ball over the top which Stockdale claimed before Rodallega could get there.
He then repeated the trick, with Rodallega this time able to make contact and hook the ball towards goal.
Stockdale caught that, but it was only moments later that Rodallega opened the scoring from a similar position, latching on to Al Habsi's long ball and calmly lifting it over the goalkeeper to put Wigan in front.
At the other end, Dempsey had the opportunity to equalise twice in quick succession but was denied firstly by a Diame challenge and then by Al Habsi's save.
Back came the hosts and Rodallega looked set to net a second, getting on the end of N'Zogbia's cross but stretching too much to control his effort.
N'Zogbia and Rodallega were soon combining again, the latter unable to finish at the near post due to the efforts of Stockdale.
Fulham boss Mark Hughes brought on Johnson, who soon had a shot blocked by Al Habsi, and the Cottagers then thought they had drawn level as Dempsey rifled in from close-range.
Replays showed the American had handled as he controlled the ball and the effort was disallowed.
Dempsey was looking threatening and saw a scissors-kick diverted wide before providing the pass that led to Fulham's leveller.
His pass found Johnson, whose strike took a deflection off Caldwell and Al Habsi's hand was not enough to stop the ball bouncing in.