The Argentina star has started the season in spectacular fashion, scoring 12 goals in his first eight appearances in all competitions.
Dybala, who has been tipped as a successor to compatriot Messi as his country's best player, scored twice against Barcelona in last season's Champions League quarter-finals but was a disappointment as Juve lost the final 4-1 to Real Madrid.
And Tardelli, who won the European Cup in 1985 to cap a glittering Juve career, says Dybala's performances in Europe remain short of the required standard.
"He's improved a lot but I still expect more from him," the World Cup-winning midfielder told Corriere dello Sport.
"There's no doubt he's a great player but, to be the new Messi, he has to lift the team in the Champions League and not only Serie A."
Juve's final defeat was their second in the last three seasons and they started this term's group stage in disappointing fashion with a 3-0 loss to Barca at Camp Nou.
Tardelli, however, is confident his old club have the strength to mount another challenge, despite their dreadful second-half showing in Catalonia.
"Getting to the final is a prestigious result but you have to win finals to get a place in history.
"I honestly didn't expect Juve to lose to Barca. You can lose to Barcelona but not without a fight, like that second half was."