The home side opened the scoring with a smashing effort from Danilo after six minutes, but a gaffe from Samir Handanovic allowed Andrea Stramaccioni's men to draw level through Wesley Sneijder only moments later.
The Dutchman combined brilliantly with Diego Milito to put Inter up for good just before the half-hour mark, before Ricky Alvarez put the match beyond Udinese's reach with a superb solo effort eight minutes before half-time.
Both sides now sit on 52 points in the league, three behind Lazio, who went down to Novara earlier in the day to breathe new life into the race for third place.
Udinese came flying out of the traps, and were ahead after only six minutes. An outrageous rabona from Antonio Di Natale allowed Pablo Armero to tee up Danilo, who cracked an effort from the edge of the box that bounced in off the post.
Kwadwo Asamoah nearly made it 2-0 within moments of the restart, but his sweetly-struck volley from Antonio Floro Flores’ flick flashed just wide.
Inter would not be down for long though. Sneijder’s shot from 20 yards out appeared to be routine for Handanovic, but the ball somehow was allowed to slip through his grasp and into the back of the net for the Dutchman’s first goal in half a year.
Sneijder’s first had an element of luck about it, but there was nothing fortuitous about his second, which arrived after 28 minutes. Diego Milito did the hard work, holding off the challenges of Danilo and Mehdi Benatia, before teeing up the Netherlands midfielder for a glorious floated finish over Handanovic.
Udinese were straight back down the pitch from the restart, with Floro Flores forcing Julio Cesar to tip his effort from 25 yards out around the post.
But the only side in the ascendancy was the Nerazzurri, and with eight minutes to go until the interval, it was 3-1. Fredy Guarin released Alvarez in the area, who then proceeded to turn Danilo inside out before slotting into the bottom corner.
The home fans were stunned into silence, but they were up in arms two minutes before half-time after seeing an apparent handball by Yuto Nagatomo go unpunished inside the area.
Inter appeared in no hurry to force the issue coming out of half-time, no doubt lured into a sense of comfort by their two-goal cushion.
Udinese dictated play, and kept the Nerazzurri penned back in their own half for large periods of the second-half. However, chances still proved hard to come by, as the visitors were able to crowd them out more often than not.
The hosts came close just after the hour mark, Julio Cesar denying Asamoah from range before getting down again to beat away Di Natale’s rebound from close range.
Sneijder went close to claiming his hat-trick with 20 minutes to go, but his effort from the edge of the box flashed just wide of Handanovic’s post.
Try as they might, the home side could not find a way through, with Julio Cesar coming up big once again to block Floro Flores from point blank range.
There was simply no way back for them, and Stramaccioni's team ran out comfortable winners in the end.
Udinese and Inter are now in fifth and sixth place respectively, three points outside of the top three with only four games to go in the Serie A season.