The Italians had trailed for much of the match after Aleksei Miranchuk gave the visitors a shock lead but, with only 13 minutes remaining, Dybala produced a wonderful equaliser, before showing his predatory instincts with a second goal shortly after to spare his side's blushes.
Another of Europe's big guns who were forced to come from behind to win were Bayern Munich, who defeated Olympiacos 3-2 in Athens.
Youssef El-Arabi put the hosts ahead midway through the first half but two goals from Robert Lewandowski either side of the break put the German champions ahead. Corentin Tolisso's strike looked to have given Bayern a comfortable win but Guilherme's deflected effort ensured a nervy end to the match.
Kylian Mbappe was the star of the show for Paris Saint-Germain, who came away from Club Brugge with a handsome 5-0 victory.
Mauro Icardi gave the visitors an early lead and the match remained tight until Mbappe's introduction on 52 minutes. Within 10 minutes, he had got himself on the scoresheet before Icardi grabbed his second of the night.
Mbappe's second was a brilliant finish after a long passing move featuring a sublime through ball from Angel di Maria, and he soon sealed his hat-trick.
Meanwhile, in the day's early match, Shakhtar Donetsk were held at home by Dinamo Zagreb.
Yevhen Konoplyanka gave the Ukrainians the advantage but goals from Dani Olmo and Mislav Orsic threatened to give Zagreb a shock win.
However, Dodo found a late equaliser to ensure a share of the spoils.