The Black Cats got some luck when home goalkeeper Marcus Bettinelli scored an embarrassing own goal to level the game after Hugo Rodallega’s opener at Craven Cottage.
Ricardo Alvarez’s fine strike and a Jordi Gomez penalty secured the Black Cats a 3-1 win and fifth round trip to giant-killing Bradford, who beat Poyet’s former club Chelsea at Stamford Bridge in round four.
And the Sunderland boss thought his side were well worth their own victory in west London.
“I was just thinking how unfair football is because we were the better team by far, we were creating and controlling the ball. We didn’t allow Fulham to play the game they like to play and we conceded a very strange goal,” he said.
“I just wanted the players to believe that something would happen. I think the equaliser was probably the turning point of the game, but overall we deserved to win the game so I don’t think it’s an action to analyse too much.
“From our point of view, it was a game we needed to get back to a certain way of football, a certain way of passing football, controlling and being aggressive.
“The result should have been a consequence of that but, in the end, something needed to happen and maybe that (Bettinelli) goal changed the game for us.
“Overall, I am pleased today. I really enjoyed it. I didn’t say this once, I think, this year. I really enjoyed watching the team today.”
Poyet is unlikely to be able to play in the same manner in the next round, given the atrocious state of Bradford’s Valley Parade pitch, with the Sunderland boss joking Phil Parkinson’s side were instead welcome to play at the Stadium of Light.
Poyet also praised the performance of Alvarez as his side progressed. The Argentina international has found opportunities restricted since arriving on a season-long loan from Inter Milan, but showed flashes of his true quality in west London.
“We hope [this is that start for him],” Poyet added. “I said the other day after the game that I have to play Ricky.
“My willingness was to find a place or make something for him to play. The circumstances gave me a chance to play him today and now everybody can see why I need him on the pitch.
“If we give him the ball in the right areas, I wouldn’t be happy if I was a full-back seeing him come every time against me with everything he has got.
“He is a great addition for us – let’s see if we can get him up to the pace of English football.
“The chance he had in Italy to recover after an action is not going to happen here, so we need to find the balance for him to start getting that rhythm.”