Vincenzo Montella's Fiorentina had failed to win their past three matches in all competitions and were looking to keep their slim hopes of a top-three finish alive at Stadio Artemio Franchi.
Diamanti had the perfect chance to put them ahead in the second half, but he fluffed his lines from 12 yards.
And Verona made them pay in the 90th minute when Neto made a hash of a routine effort from Lazaros Christodoulopoulos allowing substitute Mounir Obbadi to pounce.
Fiorentina enjoyed the better of the first-half openings and Rafael kept the scores level just before the half-hour mark with a superb save from Alberto Gilardino's near-post flick after the striker got in front of his man to meet Andrea Lazzari's cross from the left.
The flashpoint of the match came in the 67th minute when Rafael was adjudged to have brought down Gilardino, but the goalkeeper redeemed himself by diving low to his left to deny Diamanti from 12 yards.
But while one goalkeeper was hero, the other proved a zero as Neto spilled Christodoulopoulos' effort from the edge of the box and Obaddi was on hand to tap home the winner to move Verona 13 points clear of the drop zone.
'People will remember us because we got to No.1 in the world rankings and achieved great things - but our biggest regret is that we didn’t win anything': Ex-Everton star Kevin Mirallas opens up on 'honour' of being part of nation's golden generation
‘I’m sure we’d have finished in the top four or five if he hadn’t joined Chelsea in January. Him leaving damaged the team, but we couldn’t stand in his way’: Alan Curbishley reflects on failed Champions League dream after selling key player
