For much of the day, it did not appear that this FA Cup fourth-round replay would go the Black Cats' way.
Jack Rodwell pulled out with a minor knock on the morning of the game and Connor Wickham was unable to take his place in the starting line-up after picking up a calf complaint in the warm-up.
Things got worse still when Hugo Rodallega gave Fulham a 28th-minute lead against the run of play, but Sunderland at last got some luck when home goalkeeper Marcus Bettinelli scored an embarrassing own goal.
It proved to be the game-changing moment at Craven Cottage, where 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.
"I was just thinking how unfair football is because we were the better team by far, we were creating and controlling the ball," Poyet said of his emotions when his side were behind.
"We didn't allow Fulham to play the game they like to play and we conceded a very strange goal as well.
"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 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.
The Sunderland boss joked City were instead welcome to play at the Stadium of Light, after a tie in which Alvarez was the star man.
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 said
"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."
Fulham counterpart Kit Symons was understandably disappointed to exit the FA Cup on home soil, but quickly jumped to the defence of Bettinelli after he fumbled Patrick van Aanholt's deflected cross.
"Goals are generally defining moments in games," he said
"It was obviously a mistake, but Marcus has been excellent for us.
"I am certainly not going to point any blame at him
He'll be all right, he is a very strong character and he has been absolutely superb for us this season.
"One mistake like that won't affect him too bad
He is disappointed, obviously, and a bit upset, but these things happen.
"He will dust himself down and get on with it."
Source : PA
Source: PA