Poyet replaced Paolo Di Canio as manager at the Stadium of Light in September and the club will be bottom of the table on Christmas Day.
But Catttermole believes there were enough signs of encouragement from Saturday's 0-0 draw at West Ham to offer hope for the rest of the season.
"All we can do is take confidence from the game," he said.
"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.
"We have got a good squad of players. 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."