Giuseppe Sannino's side fell behind with just one minute on the clock as Maicosuel capitalised on a defensive mix-up to run through and put the visitors ahead.
But they rallied immediately, with Sergio Pellissier levelling matters 13 minutes in before Luca Rigoni put the hosts ahead just before half-time.
And the Verona outfit held on to take a victory that moves them up to 14th in the Italian top flight, while Udinese, who finished fifth last term, sit 12th.
The day's most dramatic game saw Cagliari and Sampdoria share a 2-2 draw that featured plenty of late action at Stadio Nereo Rocco.
Manolo Gabbiadini looked to have salvaged a point for Sampdoria when he cancelled out Albin Ekdal's first-half goal in the 89th minute of the match.
But Daniele Conti struck in the first minute of stoppage time to seemingly ensure that the three points would go to the hosts.
However, there was still time for one more twist, as Lorenzo De Silvestri netted in the 93rd minute to save his side from defeat in a thrilling finish.
Cagliari move up to ninth courtesy of their draw, while Sampdoria's second point of the season sees them just above the relegation zone in 17th.
In the final game of the day, Genoa held visitors Livorno to a draw that keeps the sides in 13th and sixth place respectively.
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