The midfielder was deemed surplus to requirements at the Stadium of Light during Paolo Di Canio's reign at the club, and looked likely for an exit in the January transfer window.
However, Cattermole has been restored to the side since Poyet took over, and he believes that the 25-year-old can go from strength to strength this season.
"I am sure we are going to make Lee a much better player than what people think, and that tells you how important he is for the football club," Poyet told the press.
"He is paying attention, he is always asking questions, he is always learning and if someone needed to play in this game above any game, it was Lee Cattermole, so there was no doubt in my mind about who was starting in the middle.
"Then, of course, you have got options, but I am sure he is going to be much better, and I hope he is going to keep helping us."
Sunderland moved off the bottom of the Premier League with the victory.