The Serie A champions are on track to join that trio in the Champions League quarter-finals, leading Porto 2-0 ahead of hosting the return leg on Tuesday.
Dybala, linked with the LaLiga giants previously, said Juve deserved their place alongside Barca and Madrid.
"Since the Champions League started we've been at the same level as Barcelona, Real Madrid and Bayern Munich," he said.
"If we get through, we'll also be on the same level now.
"We're also top of the league and we won the first match of the Coppa Italia [semi-final against Napoli]. In some ways, we're doing better than them."
Dybala, 23, has won three trophies at Juve – the Serie A and Coppa last season, to go with the Supercoppa Italiana.
The Argentina international has just one goal in the Champions League this campaign and while he would like to add to that tally, he has set his sights on lifting the trophy.
"It would be crazy to win the Champions League at my age because I only joined Juventus two years ago," Dybala said.
"Some of my team-mates have already won the trophy and I'd like to be decisive in a Champions League game."