Andrea Favilli produced a remarkable six-minute cameo for the visitors when he came on in the 55th minute, put Verona ahead five minutes later and then went off injured.
Juve, still missing Cristiano Ronaldo who was isolating after testing positive for COVID-19, levelled with a solo goal by Kulusevski in the 78th minute.
The Turin side, who drew their previous league game 1-1 at Crotone, are fifth with nine points after five games, three behind leaders AC Milan who host AS Roma on Monday.
They have won only once on the pitch and drawn three times while their other game was a 3-0 walkover against Napoli who failed to play due to confusion over COVID-19 rules.
Juve are clearly still at an experimental stage under coach Andrea Pirlo, who is in his first senior coaching job, and Verona were the more dangerous side for the first hour.
However, Kulusevski brought them to life and they laid siege to the Verona goal in the closing stages.