Poyet replaced Paolo Di Canio at the Stadium of Light in September but, despite improvements in performance, Sunderland will be bottom of the Barclays Premier League on Christmas Day after failing to find a winning goal despite creating the better chances in Saturday's 0-0 draw at fellow strugglers West Ham.
The former Brighton boss said after the game that it was the first time he had seen 'his' Sunderland side play and Cattermole revealed that Poyet believes similar displays for the remainder of the campaign will see the Black Cats safe.
"All we can do is take confidence from that game," said Cattermole.
"The manager said it is the best performance he has seen from his Sunderland side - we controlled the game throughout. The message from the manager is that if we play like that for the rest of the season we will have no worries."
Former Wigan midfielder Cattermole was making his first start since being sent off in a 1-0 defeat at Hull on November 2 and the 25-year-old feels that the strength in depth required to survive a relegation battle is evident at Sunderland.
"We have got a good squad of players," he added.
"Obviously we are bottom of the league but there is a lot of quality. The gaffer has rotated quite a bit and I think he is just trying to find his best XI.
"We have got some great players on the bench today (Saturday), we have just got to keep going and the lads have got to be patient like I have been."