The hosts went ahead in the 57th minute through Hugo Rodallega, who neatly guided the ball over goalkeeper 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 Barclays Premier League drop zone up 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 had the first attempt on goal after eight minutes, striking a low shot from outside of the area that Al Habsi got down to his left to save comfortably.
Two minutes later, Wigan launched an attack that 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 close 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 player.
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 that 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 that 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 a few 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 he 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 when he got on the end of N'Zogbia's cross, but he stretched too much to control his effort.
N'Zogbia and Rodallega were soon combining again, the latter unable to apply the 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.
The effort was disallowed though, and replays showed the American had handled as he controlled the ball.
Dempsey was looking threatening and he had a scissor-kick diverted wide before providing the assist that led to Fulham's leveller.
His pass found Johnson, whose strike took a deflection off Caldwell, and Al Habsi could not stop the ball bouncing in, despite getting a hand to it.
Source: DSG
Source: DSG