The Middlesbrough academy graduate has had a hand in eight goals in his last four Premier League appearances and scored in Saturday's 3-0 win at local rivals Newcastle United.
Poyet believes that too much was expected of Johnson after a £10 million move from Manchester City last season, and is not surprised to see the England international flourish with more support from his team-mates.
The Uruguayan said: "The game is not just about passing the ball, it is about doing the basics all the time, and in the beginning we were not doing it, but now they are confident they can do that all the time, so whatever team we pick everyone knows what to do.
"An example is Adam Johnson.
"For some reason at this football club we were giving him the responsibility of doing everything, thinking you come from Man City so give him the ball let him take on 10 players and score every game, but it doesn't work like that.
"We have to play in a way to help Adam and he is doing it every week, it is a bit of responsibility for Adam and a bit of the team."
Another former Middlesbrough player at the Stadium of Light is Lee Cattermole, who remains in the squad after a deadline-day move to Stoke City collapsed.
Cattermole did not feature in the derby win due to a calf injury, with replacement Liam Bridcutt - a deadline-day arrival from Brighton and Hove Albion excelling.
But the tenacious midfielder remains firmly in Poyet's plans, who is happy to have kept hold of the club captain.
He explained: "I am delighted that the chairman said to me, 'If you need him he stays'. It is about having the players you need to stay up and Lee is one of those.
"We did not want to lose any of our main players. Sometimes a club might put an offer that makes you think, you like to say everyone has a price. I say sometimes you have to be realistic but I am delighted we don't lose any of our most important players."
