Parma are facing uncertain future, with concerns that they may not even be able to finish the season after being declared bankrupt and entering administration last month.
On-loan winger Varela finally gave their supporters something to shout about when his attempted cross found the far corner of the net 20 minutes from time in front of a sparse crowd at Stadio Ennio Tardini.
Varela's first goal for the club ensured Roberto Donadoni's men sealed their first victory in 11 games in all competitions and their only Serie A success since seeing off Fiorentina back in January.
Rock-bottom Parma must now hope for new investment to provide a financial lifeline and save the club from going out of business.
Udinese defender Igor Bubnjic came close to opening the scoring when he hit the post in the first half following a mistake from Parma goalkeeper Antonio Mirante, but Andrea Stramaccioni's side are now without a win in four Serie A matches.
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