Prior to arriving in London, Sunderland had won just once away from home, and three times in total, but Poyet believes something clicked into place for the Black Cats as they raced away with three points.
Sunderland have been entrenched in the bottom three since losing to Fulham on the opening day of the season but Poyet was delighted with his side's performance as they moved ahead of Crystal Palace on goal difference.
"Something happened at the right time today," he told Sky Sports. "We needed to perform, it was time to go and do something special and the lads were superb.
"I think we played a fantastic performance to score four goals away from home, it came at the right time against one of the teams that are around in the relegation battle."
Johnson opened the scoring on the half-hour mark with a venomous free-kick before doubling his tally in the 69th minute.
The perfect afternoon was sealed for the winger, who terrorised Fulham throughout, when he slotted home a penalty deep into the second half and Poyet believes Johnson's display was second to none.
"It was as good as you can get," he said. "We always say how special it is to score a goal or two but to score a hat-trick away from home, in the situation that we are, there are not many who could.
"We know the quality of [Adam] Johnson and we needed to help him. We needed to play a way that he could enjoy and go on to produce the difference.
"He's been doing it the past three games so it's important that we keep him happy so he can perform."
Johnson turned provider for Sunderland's second, playing a smart training ground move from a free-kick to set Ki Sung-yueng up for his first open-play Premier League goal.
Sunderland have scored eight of their 19 goals from set-pieces and Poyet admits there has been a lot of work put in on the training ground.
"We've been doing different things and it's a combination of practicing, players getting confidence and choosing the right option at the right time," he added.
It was not all plain sailing for Sunderland however as Fulham scored straight after the break to give themselves a lifeline at 2-1.
However, a fantastic piece of counter-attacking football, which saw Ki race into Fulham territory before sliding a ball into the path of Johnson, reasserted Sunderland's control and Poyet believes it could not have come at a better time.
"It came at the right time," he admitted. "In football, when you're 2-0 up and then the home team come back, we had a couple of breaks but couldn't finish them off.
"But Johnson, it was very easy for him today. We got what we deserved, we were the better team by far.
"[Getting off the bottom of the table] is one of the key points to get out of today.
"With the table so close, so many people around us, to score the goals that we did, it's an honest top job. "