Inter dominated the first hour and missed a flurry of chances including one incredible effort from Brazilian Jonathan who contrived to fire wide from six metres out in the 54th minute with the goal at his mercy.
Rodrigo Palacio was also guilty when he slipped and fired wildly over when he had a clear run on goal.
Di Natale made them pay in the 63rd minute when he ran onto Andrea Lazzarri's chipped pass to score with a diagonal shot and put Udinese ahead.
Inter had Brazilian defender Juan Jesus dismissed two minutes later for a second bookable office before Luis Muriel and Di Natale, who took his Serie A tally to 12 goals, struck twice in a four-minute spell.
Serie A leaders Juventus slumped to a 2-1 defeat at home to lowly Sampdoria who came from behind to win despite playing for an hour with 10 men.
Juve appeared to be set for a routine win when Sebastian Giovinco put them ahead with a 24th minute penalty and Sampdoria had Gaetano Berardi dismissed for a second bookable offence seven minutes later.
Instead Mauro Icardi scored twice in the second half as Sampdoria hit back.
Third-placed Fiorentina also went down to a surprise home defeat, losing 2-0 to relegation-threatened Pescara.
Juventus, with 44 points from 19 games, still reached the halfway mark in the championship with a five-point lead over Lazio, who beat Cagliari 2-1 on Saturday. Fiorentina and Inter have 35 points.
In other matches, two goals by Marco Boriello gave Genoa a 2-0 win over Bologna which lifted them out of the relegation zone while Bojan Krkic and Giampaolo Pazzini scored to give seventh-placed AC Milan a 2-1 win at home to bottom club Siena.
Chievo Verona beat Atalanta 1-0 in a midtable clash and a stoppage-time goal from Amauri gave Parma a 2-1 win over Palermo which left the Sicilians one place off the bottom with 15 points, one below Cagliari and two adrift of 17th-placed Genoa.
Igor Budan had put Palermo level in the 85th minute, wiping out Ishak Belfodil's opening goal.
Fiorentina should have been several goals clear at halftime against Pescara but Perin brilliantly kept the hosts at bay with saves from Alberto Aquilani, twice, and Borja Valero.
Jonathas then headed the visitors in front after 57 minutes and, despite further Fiorentina pressure, Celik added another in stoppage time when he won the ball off Sebastian Rocaglia and slid his finish under goalkeeper Neto.