Massimilano Allegri's team jumped one point above AC Milan and to within just four of league leaders Napoli thanks to a beautifully-constructed goal finished off by the Brazilian defender four minutes from the end at the Allianz Stadium.
Danilo extended Juve's league winning streak to eight games, a run in which they haven't conceded a goal, after Federico Chiesa controlled fellow substitute Leandro Paredes' delicate chip and then played a perfect pass across goal.
That was one of the few clear-cut chances Juve created against eighth-placed Udinese, who were missing their talisman Gerard Deulofeu and are winless in nine matches.
On Friday, Juve travel to Napoli, who on Sunday are in action at Sampdoria, where former Italy forward Vialli -- a victim of pancreatic cancer -- played most of his football and may be mourned the most.
"After the win our first thoughts go to Gianluca Vialli. He was a great guy, I was lucky enough to have had him in my life," said Italy forward Chiesa.
The 25-year-old was part of the Italy team which won Euro 2020, and hailed Vialli's role in that triumph despite him not having a coaching role with the Azzurri set-up.
"You could speak for hours and hours about what a great man he was," Chiesa added.
"This win is for him"
It was an emotional day for many Juve fans, who will remember Vialli as the last captain to lead their team to a major European honour more than a quarter of a century ago.
Fans brought jerseys emblazoned with the former Italy striker's name, while one flag in the Allianz Stadium stands depicted the moment he lifted the Champions League trophy in 1996.
Gianluca Pessotto, who played over 250 times for Juventus and was alongside Vialli in that triumphant campaign, hailed "a captain. A friend," in his pre-match eulogy on the pitch.
"You were our leader on and off the field," said former Italy international Pessotto who is now sporting director of Juve's youth academy.
"Nobody will ever forget the memories you gave us with your football and your goals. We will miss your smile so much."
Juve were also mourning former Italy defender Ernesto Castano, who won three Serie A titles with the Turin giants in the 1960s and died aged 83 on Thursday.
Castano won the 1968 European Championship with his country and was a founding member of the Italian Footballers Association alongside the likes of Gianni Rivera.
Also bidding a farewell of sorts was Andrea Agnelli, at his final home game as Juve chairman after announcing his resignation late last year with the club accused by prosecutors in Turin of a variety of financial crimes.
Agnelli, along with the rest of the club's board -- including former star midfielder Pavel Nedved -- will be replaced later this month but he still had time to hand Massimiliano Allegri the manager of the month award before kick-off.
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