Martin Rueda's men stunned Udinese on home soil 3-1 two weeks ago, with Bobadilla's treble sealing their first points in Group A.
And the Argentinian was on target to set the Swiss outfit on their way before veteran striker Antonio Di Natale levelled early in the second half - seconds after he had missed from the penalty spot.
However, the visitors re-established their lead through Farnerud and Nuzzolo before Diego Fabbrini set up a tense finale.
The result sent Young Boys up to third in the group, level on six points alongside Liverpool, while Udinese prop up the table.
Head coach Francesco Guidolin had been forced to defend his side's attitude towards the competition after their defeat in Bern two weeks ago.
They started in promising fashion tonight, with Di Natale rattling the bar from close range before the former Italy international struck narrowly wide.
Yet they fell behind in the 27th minute, with Bobadilla proving to be Udinese's nemesis once again.
After being fouled by Larangera Danilo just outside the area, the former Borussia Monchengladbach striker drilled his free-kick past Udinese keeper Zeljko Brkic.
Young Boys defended stoutly until the interval but seemed to have been caught napping as, seconds into the second half, Math Ranegie was fouled in the area by defender Alain Nef.
Di Natale's low effort, however, saw his low effort saved by Marco Wolfli but the 35-year-old would make amends almost immediately, volleying into the net after the Young Boys keeper could only clear Pablo Armero's cross into his path.
Wolfli saved well at the feet of Ranegie but, at the other end, Farnerud curled wonderfully into the bottom corner from the edge of the box to give Young Boys the lead once again in the 65th minute.
The Bern-based outfit then left the Stadio Friuli faithful stunned seven minutes later as Nuzzolo slotted home after Brkic could only parry Gonzalo Zarate's effort from a narrow angle.
To their credit, the Bianconeri, who had not been beaten in their last seven home European matches prior to tonight, refused to lie down and halved the deficit when Fabbrini capitalised on some loose defending to prod home.
Ranegie's strike was tipped over by Wolfli as Udinese sensed an equaliser but Young Boys held on to secure another famous victory.