Modest Udinese sold three of their best players before the season and never directly replaced them yet they could have been sitting in the Champions League qualifying round berth with just 10 games to go if they had weathered Napoli's late storm.
Midfielder Giampiero Pinzi grabbed a rare goal to put the hosts ahead in the first half in Udine before evergreen top scorer Antonio Di Natale hooked the ball over the line after Pinzi's second-half header had come off the post.
Udinese's Diego Fabbrini received a second booking in the 60th minute before Cavani missed an awful penalty and then scored a sublime free-kick for Napoli, knocked out of the Champions League last 16 by Chelsea in midweek.
The Uruguay striker then completed the comeback from a tight angle five minutes from time.
Lazio stayed a point above the pair after they lost 1-0 at Catania while Inter Milan are struggling to stay in contention for the Champions League qualifying place after a 0-0 home draw with Atalanta.
Former Juventus defender Nicola Legrottaglie grabbed Catania's winner following a corner 11 minutes from time to shoot Vincenzo Montella's overachieving Sicilians up to joint-sixth.
"Now we can target new objectives, maybe even aiming at a European place," former Roma boss Montella told reporters.
"But let's leave the dreams to the fans. We have to stay realistic."
Lazio's defeat also meant leaders AC Milan and second-placed Juventus, who won 2-0 at Parma and 5-0 at Fiorentina respectively on Saturday, have almost unassailable cushions over the rest in the two automatic Champions League places.
Milan are top with 60 points from 28 games followed by Juve on 56 and Lazio on 48. Udinese and Napoli - in two of three potential Europa League berths - are six clear of Inter, Catania and AS Roma, who host Genoa on Monday.
Inter, dumped out of the Champions League by Olympique Marseille in midweek, have little hope of making next season's competition after Diego Milito missed a first-half penalty.
Coach Claudio Ranieri denied reports striker Diego Forlan refused to come on as a substitute.
"I was talking to him about tactics," he said.
Mauricio Pinilla bagged a hat-trick as Cagliari coach Massimo Ficcadenti - reappointed last weekend four months after being sacked - returned for a 3-0 win over the nine men of ex-club Cesena, who are rock bottom and 14 points from safety.
Second-bottom Novara won 2-0 at Siena to boost their outside chances of avoiding the drop and Lecce - in the third and final relegation spot - drew 1-1 at home to Palermo to stand five points adrift of Parma, who are 17th in the 20-team standings.
Chievo drew 2-2 at fellow mid-table side Bologna.
