The Friuli outfit visit Atalanta on Sunday looking to extend their five-game unbeaten run.
And Di Natale, who got on the scoresheet in his team's 2-0 triumph over Bologna last time out, believes his side cannot get carried away.
"We have done really well in our opening five games," the 34-year-old Di Natale said.
"But it's too early to know how far we will go this season.
"We know that it takes very little for things to change and that is why we must continue to work hard, focus on ourselves and not on others."
Udinese are level on 11 points with Juventus heading into Sunday's contest in Bergamo.
Francesco Guidolin's men have shown great balance at both ends of the pitch with seven goals scored and just one conceded in their past five games.
Atalanta, meanwhile, will try to bounce back from their 3-1 loss at Roma in the last round.
That defeat was Atalanta's first of the campaign and ended a three-match winning streak for the Bergamaschi.
Had Atalanta not been punished with a six-point penalty for a betting scandal at the start of the campaign, Stefano Colantuono's side would be in the top half of the standings.
As it is, the promoted outfit go into Sunday's match in 16th place but unbeaten at the Atleti Azzurri d'Italia stadium.
"We want to do well following our defeat at Roma," Colantuono said.
"On Sunday we know we face a challenge against Udinese.
"Udinese are very strong and are playing very well."