The combative midfielder was deemed surplus to requirements by former Black Cats boss Paolo Di Canio, with a January exit from the Stadium of Light all but guaranteed.
He has, however, been drafted back into the fold in the wake of Di Canio's departure and has responded with a string of typically robust performances.
Cattermole was a key cog in the Sunderland engine room again on Sunday as they collected their first win of the season in a crucial Wear-Tyne derby clash with Newcastle United, and Poyet is delighted to have the 25-year-old at his disposal.
The Uruguayan said: "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.
"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."
Victory over the Magpies came in Poyet's second game in charge, and he admits emotions were running high when substitute Fabio Borini crashed home a dramatic winner.
He added: "I think I am going to sleep well tonight, I am going to sleep very well tonight. It's going to be like I played the game myself, I chased the ball, I celebrated.
"I was close to running to the corner, but I was trying to tell a few players what to do, so I didn't know which was the most important one.
"Yes, it's a lot. Some people don't realise, it's incredible."