Wilson came on in the 82nd minute and equalised in the second minute of injury time by meeting a cross from Adama Traore and, with his back towards the goal, flicking the ball inside the far post.
He netted the winner four minutes later with a stooping header from another cross from Antonee Robinson as Fulham were finally rewarded for their attacking play.
Fulham had only been behind for 113 minutes in the league all season and controlled most of the early action, only for Brentford to take the lead in the 24th minute against the run of play.
Sloppy play in midfield by Fulham gave the visitors possession and German midfielder Vitaly Janelt let fly with a shot from almost 30 metres out that flew inside Bernd Leno's left-hand post.
Brentford's Vitaly Janelt (front) celebrates after scoring against Fulham. (AP PHOTO)
Marco Silva's Fulham side pushed forward but never looked like scoring until Wilson's late double.
The win lifts Fulham into ninth place in the league table, while Brentford remain in 13th spot.
"It's emotionally very tough right now," said Brentford coach Thomas Frank.
"When you're leading in the 92nd minute, it's tough for us to even have a draw - and then we lose it.
"On the day, Fulham were better, but we showed a lot of resilience and defending.
"I thought that could give us the win, but then there was an unbelievable touch from Wilson where 99 times out of 100 it doesn't go in. Then we could have won, and in the end, we lost it. It was brutal."
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