Sunderland face Aston Villa on New Year's Day and can move out of the bottom three with a win, which would also extend their Premier League unbeaten run to five games.
Poyet is delighted with his side's recent performances, but knows there is still a long way to go in their battle for survival.
"A few weeks or a few months ago, people were looking at the table and thinking it was impossible," Poyet said.
"And now we are there, so we have to make sure we don't stop and don't get complacent because we have done nothing."
Despite the warning, the Uruguayan praised his side for the timing of their revival and feels his side are building a pivotal rapport with the club's fans.
"I think it (the unbeaten run) has come at a great time," Poyet said.
"I've always believed that Christmas is a key moment (because there are) so many games.
"I think the connection between the fans and the players is paying dividends; it's giving everybody that extra bit of energy and belief."
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘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
