Two goals in two minutes from Erik Lamela appeared to have put the Giallorossi on the way to victory, but they suddenly stopped playing, allowing Udinese to hit back at them.
Maurizio Domizzi pulled a goal back before half-time and Antonio Di Natale brought them level early in the second half.
Di Natale then scored the winner from the penalty spot and Panagiotis Tachtsidis completed Roma's self-destruction by earning a red card in stoppage time.
The tactics of Zdenek Zeman do not leave much up to the imagination - either his side are attacking and scoring goals, or they are conceding by the bucket load.
The first half an hour at the Stadio Olimpico showed the best side of Zeman's tactics, and they duly opened up a two-goal lead.
Lamela scored the first in the 22nd minute when he picked up the ball to the right of the goal on the goal line and a path towards goal opened up in front of him.
All he had to do was lift the ball in from a tight angle after Zeljko Brkic committed too early to him crossing the ball, leaving his inside post unguarded.
Two minutes later, Francesco Totti sent through Osvaldo, whose cross was headed in from eight yards by Lamela.
After that, though, Roma capitulated and enabled Udinese to get back into the game.
Domizzi pulled a goal back in the 32nd minute when Osvaldo inadvertently set up Domizzi, who accepted the gift to beat Maarten Stekelenburg inside the far post.
The visitors almost levelled five minutes later when a fine move of one-touch football culminated in Di Natale shooting just wide of the far post.
Di Natale brought Udinese level in the 50th minute with Roma failing to clear their lines after the striker's first shot was saved by Stekelenburg.
The ball eventually found its way to Pablo Armero, who's shot was turned in by Di Natale after he swivelled on the spot, striking the ball with the outside of his boot to fool Stekelenburg.
Roma surged forwards again, but they were caught out on the counter-attack with Castan committing a foul inside the penalty area.
Di Natale stepped up and stole Totti's trademark to chip his spot-kick in and earn Francesco Guidolin his first ever win in the capital.