The Italian champions drew 2-2 at home with Galatasaray on Wednesday night having previously been held 1-1 at Copenhagen in their Group B opener.
With back-to-back games against nine-time European Champions Real Madrid awaiting them next, as well as a difficult trip to Turkey, Conte is not brimming with optimism.
"Now our journey is hard," Conte told Gazzetta dello Sport.
"But this should not discourage us. We have to continue to play with a knife between our teeth."
Conte had reason to be disillusioned on Wednesday night as Juve rallied from 1-0 down to take a 2-1 lead in the 87th minute.
However, Gala substitute Umut Bulut equalised one minute later to silence the home crowd.
"It's disappointing to concede a goal immediately after going ahead," Conte said.
"We need to be more careful, especially in this competition.
"We did do a good job to come from behind and take the lead but in the end my players looked tired."
The result has left Juve in second place, four points behind pool leaders Real Madrid and one point clear of Copenhagen and Galatasaray.
Juve travel to the Bernabeu on October 23 before hosting Real Madrid on November 5.
The Bianconeri take on Copenhagen in Turin on November 27 before ending the group stages against Galatasaray in Turkey on December 10.