The record 36-time champions trail second-placed Inter Milan on goal difference after Maduka Okoye's own goal and Nicola Savona's effort.
Their next game is Tuesday's Champions League trip to Lille with Thiago Motta's Italian giants down in 14th place in the league phase table.
Juve were without a win since October 19 after draws with Inter Milan and Parma as coach Motta brought in-form Turkey teenager Kenan Yildiz into the starting lineup.
Udinese had lost just only one home game this season against Inter in late-September and were without defender Isaak Toure following his red card in the midweek defeat at Venezia.
The visitors took the lead after 20 minutes as Udinese goalkeeper Okoye unluckily turned the ball into his own net after a Marcus Thuram shot hit the post.
Two minutes later the hosts had a golden chance to equalise but English striker Keinan Davis had his effort from inside the box blocked.
Juve then doubled their advantage after 37 minutes as right-back Savona fired home a re-bound from 19-year-old Yildiz's shot which also hit the woodwork.
Udinese stayed in the game and were denied the chance to halve the deficit as Davis' 52nd minute effort was disallowed for a foul.
Later AC Milan can go above Udinese if they win at Monza, who are owned by their former president Silvio Berlusconi and coached by their former defender Alessandro Nesta.
Earlier, Italy winger Riccardo Orsolini scored for a second time in five days as Bologna edged lowly Lecce 1-0 to move up to eighth place.
Their third domestic victory of the season came before they play Monaco in the Champions League on Tuesday.
"There are still things to improve," Bologna coach Vincenzo Italiano said. "This team has quality and is on the right track."
On Sunday, league leaders Napoli host fourth-placed Atalanta before Inter Milan in second welcome struggling Venezia.
