Juve could have taken the lead inside the opening two minutes when Dybala found space inside the penalty area, only to fire his effort inches past the post.
Mario Mandzukic also came close as he continued his search for a first league goal of the season, but having latched on to a rebound in the 26th minute he proceeded to blaze his finish over the crossbar.
That wastefulness was punished on the half-hour mark as Udinese broke the deadlock through Jankto, who fired home from the edge of the penalty area.
The lead should have lasted only four minutes as Juventus immediately went in search of an equaliser, but Stephan Lichtsteiner somehow put the ball wide from inside the six-yard box.
Juve would not have to wait too much longer to restore parity, however, with Dybala doing so in style shortly before half time with an exquisite free kick that left the keeper helpless.
It was another dead-ball situation that handed Dybala his second of the night too, completing the turnaround from the penalty spot six minutes into the second half.
Rodrigo De Paul was penalised for the visitors, and Dybala made no mistake by picking out the bottom corner from 12 yards.
The champions held out for the win, though, recording their 19th consecutive home Serie A win.