The Red Devils signed Uruguay legend Cavani on a free transfer in October following his departure from PSG at the end of last season.
Cavani scored 200 goals in 301 appearances in Paris and will hope to enjoy more success at Old Trafford.
The 33-year-old came off the bench to score as Manchester United beat Everton 3-1 before the international break and Poyet says this was further evidence that he can have a big role at the club.
The former Chelsea and Tottenham star told MailSport: ‘I think he is going to be more important for Manchester United than people think.
‘Look at the last game at Everton. The result was in the balance, he came off the bench, once chance and bang.
‘English football is perfect for Uruguayans because we are passionate. Once we sign a contract, we forget about it because the only thing that matters next is winning. English fans love that.
‘He makes an incredible amount of runs. Maybe he makes an unbelievable run and the ball isn’t played.
‘The fitter he gets, the more he understands United’s play, the more he will play.’
Manchester United captain Harry Maguire also talked up Cavani’s influence after he scored his first goal for the club.
‘I think he’s going to be a big player for us, he’s going to score goals. He knows where the back of the net is and I’ve seen that in training a few times already.
‘He’s just getting up to speed in terms of his training and work rate but he’s going to be a big player for us.’