The much-travelled former France striker joined the Italian giants in January on a five-month deal, ending a one-year spell with China's Shanghai Shenhua.
Juventus, who won the European Cup in 1985 and 1996, are three points clear of second-placed Napoli at the top of Serie A and will travel to Celtic in a Champions League last 16 first leg tie on February 12.
"It is a very difficult competition," Anelka told Juventus TV. "When you start out you never know if you'll be able to win it.
"I believe a team like Juventus have the players, mentality and athletic qualities required to do so. We'll give all our strength and energy to bring the trophy home."
The former Chelsea, Arsenal, Real Madrid and Manchester City forward said he had already been made to feel at home at the Turin team.
"I've joined a great club with great players," he explained. "I'm happy to be here... and very proud to be able to wear this shirt.
"The players have told me it's like a family and when I arrived I got that feeling straight away as I walked in the dressing room.
"Italian football is renowned all over the world and I know it's a very difficult league. I've never played here and not come up against many Italian teams but I know it's really tough," added Anelka.
"For my style of play and mentality the English league was the best, however it's been good to have the different experiences in my career. I've developed and had the opportunity to appreciate different atmospheres and ways of life."
Anelka left Chelsea for China in the 2012 January transfer window and was later joined by his former Stamford Bridge team-mate Didier Drogba.
Both have now moved on though, with Drogba having sealed a switch to Turkish side Galatasaray earlier this week.
"China was good," said Anelka. "I lived in a magnificent city but it was a slightly complicated experience as far as their approach to the game was concerned.
"But I still appreciated it and have no regrets."
