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."
Following Steve Clarke's sacking from West Brom on Saturday evening, Sam Allardyce is the only manager of a club in the bottom five who was in charge at the beginning of the season.
He saw his side pass up the opportunity to beat the bottom side at home and put some breathing space between the Hammers and the relegation zone.
But Allardyce, who is still waiting to unleash club-record signing Andy Carroll as the England striker works his way back from a heel injury, knows where his side needs to improve.
"It wouldn't have mattered how we played - good, bad or indifferent - if we'd got the points, but we couldn't turn it around and find that bit of quality when needed," he said.
"On the other side of it, our magnificent defence was there again and it's the backbone of our team. That is eight cleans sheets in 16 matches. We only got 11 all season last year and finished 10th, so nobody needs to tell me what the problem is and always has been - scoring a goal."
Only Cardiff and Sunderland have scored fewer than the 13 goals West Ham have bagged so far this year and Allardyce knows the slim margins of victory can be the difference between survival and relegation.
"It's not happening for us at the minute," he added.
"We got one side of it right and now we need to keep pressing on to get the other side right in terms of scoring goals. If we score and get a clean sheet, then we'll win games.
"We've only won three of our eight clean sheets and we only needed to put another two on top of that and we'd be in the top ten and not the bottom four. That's how close it is.
"We're small margins away from getting up that table. We'd have all loved three points, but in the end I thought it was a fair result."