Arturo Vidal put Juve ahead from the spot shortly before half-time, but Real hit back seven minutes after the break through Cristiano Ronaldo.
Gareth Bale then put the Spanish side ahead on the hour mark before Fernando Llorente salvaged a point for the Serie A champions.
The result leaves Juve propping up their group with just three points and no wins, but Conte insists that the competition is not over for his side.
"We can play at their (Real's) level, which we had already proved in Madrid two weeks ago," said the Juve coach. "We will still have something to say in this competition.
"We feel stronger after this game, and we know what we do on the pitch decides who is going to win or lose, it's up to us."
Goalkeeper Gianluigi Buffon feels the Italians should be proud of their point but admits qualification from Group B is now an uphill task.
"It's no disgrace to drop points against a big team like Real Madrid," he said. "They have great individual quality, wonderful strikers and every time they get near the goal you can feel the pressure rising.
"It won't be easy for us now. but we'll do what we can to qualify."
The veteran goalkeeper reserved special praise for his opposite number, Iker Casillas, who made a string of crucial saves as he was restored to the first team.
"Iker Casillas did really well," he continued. "His saves helped his team to get a draw."