Juve have swept all before them in Serie A in recent years, winning three straight titles under former coach Antonio Conte.
They have not been able to replicate that success in Europe's premier club competition, though, following up a quarter-final exit in the 2012-13 season by failing to make it out of the group stage last term.
Juve were drawn alongside Atletico Madrid, Malmo and Olympiacos in their Champions League group this season and they are favoured to reach the last 16.
But Barzagli – who acknowledged just how much of a priority success in Europe is – will not be taking Juventus' Group A opponents lightly.
"What we hope for is to at least progress to the next stage," Barzagli said in a question and answer session with fans.
"Last year was a big disappointment for all, especially in the Champions League two years ago when we were on a good path and it was ended against Bayern, who then went on to win the competition.
"Certainly we hope to do better, we want to do better and now we have to have the mentality to progress.
"However, it will not be easy because the Champions League is always a very tough competition.
"Certainly there are trickier teams [that are in other groups], whilst even last year it looked like a comfortable group, but in European competition you cannot trust a team that on paper may seem beneath you."
Juve open their Champions League campaign with a home clash against Malmo on September 16.
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