Poyet's men put in an improved performance following last week's 8-0 humbling at Southampton when hosting Arsenal at the Stadium of Light on Saturday, only to be undone by individual errors.
Wes Brown and Vito Mannone were at fault for both goals, their mistakes gifting Alexis Sanchez a double.
Poyet now has nine days to get his men back on track as Sunderland attempt to add to their solitary Premier League victory this term when they tackle Crystal Palace a week on Monday.
The manager said: "It is a long week because we play on Monday but lifting the players is part of the job.
"We are getting paid to win games and we have won just one. It is time to win again."
Mannone, in particular, will come in for criticism having struggled at St Mary's and then dawdled on the ball on Saturday to allow Alexis to wrap up the result late on.
Poyet, though, refused to be drawn on the goalkeeper's form, saying: "I will be assessing everybody's mood. I think that is important."
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
‘He’ll still be playing at 40 at a good level because he’s in such good shape and looks after himself so well. He does everything to be a top professional’: Ex-Liverpool coach insists Mohamed Salah has plenty more miles in the tank
