Cattermole believes that Poyet and the team can take encouragement from their performance against the Toffees.
"We definitely still believe in the manager," he told The Journal. "It has been a really strange season, but we'll come to all that at the end of it.
"But the lads fully back the manager, training has been good and anybody at the stadium today would see that the fans have backed us all the way. We're very grateful.
"That's the first time I've played on the losing side at Sunderland and not been booed, so it was a sign of appreciation at the performance I thought."
Sunderland's next match is at Manchester City on Wednesday evening.