Three points were a must for the home side, who move into second place in the standings with one game of their season left to play.
The Whites, now unbeaten in 23 league matches, are two points in front of Cardiff City ahead of the Bluebirds' game in hand at Hull City on Saturday.
League One-bound Sunderland gave a good account of themselves, edging in front through Joel Asoro half and hour in, only for Lucas Piazon and star man Aleksandar Mitrovic to turn the game around for Fulham.
With the pressure off following last week's relegation, the Black Cats fired away the game's first shot 12 minutes in through Asoro.
That attempt deflected wide, while Ashley Fletcher sliced over the bar soon after as the visitors looked to make it four away games without defeat.
A big moment in the match arrived shortly before half time as, after a couple of penalty appeals went against Sunderland - one for a challenge by Tim Ream on Ovie Ejaria - Fulham were back on level terms.
Ryan Fredericks's cross was punched to the far post by Jason Steele, where Piazon was waiting to pounce - the sixth time the attacking midfielder has been directly involved in a goal at Craven Cottage in his last eight appearances.
Johansen and Aboubakar Kamara were next to try their luck, the latter forcing Steele into a save, before the visiting keeper got across to deny Mitrovic from the Serbian's first sight of goal.
Fulham were turning the screw as the final 30 minutes approached, with Cairney the next to be denied by Steele and Kevin McDonald shooting over.
Just when it was looking like being one of those days for the Cottagers, a marginally offside Mitrovic got on the end of Johansen's cross to guide the ball past Steele with a downward header.
Fulham, who boast the most Championship goals in the final 15 minutes of games, nearly added to their tally through Johansen's well-taken shot.
A one-goal win was all that mattered for Slavisa Jokanovic's charges, though, piling the pressure on Cardiff ahead of their meeting with Hull.
FULHAM (4-3-3): Bettinelli; Fredericks, Odoi, Ream, Targett (Kamara 40'); Cairney (Kalas 90'), McDonald, Johansen; Piazon (Fonte 74'), Mitrovic, Sessegnon
SUNDERLAND (4-2-3-1): Steele; Jones (Matthews 46'), Kone, O'Shea, Wilson; McNair, Robson (Maja 87'); Asoro, Ejaria (McManaman 77'), Honeyman; Fletcher