The Bianconeri took an early lead on Wednesday evening through summer signing Mirko Vucinic, before the Montenegrin turned villain by getting sent off on the stroke of half-time.
And Bologna immediately capitalised on their numerical advantage by equalising through Daniele Portanova to earn a share of the spoils.
But despite missing an opportunity to claim an outright lead at the top of the Serie A table by making it three wins from three, Barzagli believes Juve still have plenty to be happy about.
"It's only the third game of the season, but we have already seen that this year we have something more [compared to last season]," Barzagli said, according to Tuttosport.
And while the Italian international believes the Bianconeri should be winnings games like the one on Wednesday, he conceded that Bologna had been worthy opponents on the night.
"We must continue to improve to win games like the one against Bologna.
"But I think that Wednesday night there was a lot of bad luck, and their goalkeeper made some great saves."
Juventus remain joint-top in the Serie A alongside Genoa and Udinese on seven points, and will be looking to maintain their unbeaten start to the season on Sunday when they travel to Catania.