AZ led 2-0 from the first leg but saw that advantage wiped out in a nightmare opening 15 minutes where Antonio Di Natale netted twice for Udinese and the visitors reduced to 10 men following Nick Viergever's early sending off.
However, just as things were looking bleak for the Dutch side, Erik Falkenburg made it 2-1 in the 31st minute to give his side a priceless away goal and leave Udinese needing to net twice more to go through.
That proved to be beyond the hosts, who could not make the most of their man advantage or the second-half penalty miss by AZ midfielder Rasmus Elm as they exited the competition.
Needing something special to overturn their first-leg deficit, Udinese could hardly have made a better start as in their first serious attack Antonio Floro Flores was brought down in the box by Viergever.
The AZ defender was shown a straight red card for the foul and Di Natale slotted in the resulting penalty to give Udinese a massive boost.
The home side then needed just 13 more minutes to draw level on aggregate following a fine team goal.
A quick passing move around the AZ area ended up with Giovanni Pasquale being played in behind the visiting defence and his first-time ball across goal was volleyed in at the far post by Di Natale.
At that point everything looked to be going Udinese's way, but they suffered what would prove to be a decisive blow just after the half-hour as AZ pulled a goal back.
Falkenburg picked up possession inside a congested penalty area before neatly side-stepping an outrushing defender and firing a left-footed shot into the top corner.
Udinese tried their best to recover from that setback, but Floro Flores and Di Natale could not make the most of their opportunities before the break.
The home side continued to push after the interval, but their hopes should have been ended in the 63rd minute when Brett Holman went crashing in the area following a challenge by Maurizio Domizzi to earn AZ a penalty.
However, the opportunity was badly wasted as Elm smashed his spot-kick over the bar.
AZ were then denied by a superb double save by Samir Handanovic, who kept out efforts from Jozy Altidore and Holman, as Udinese just about kept their heads above water - but they could not capitalise as they bid farewell to Europe.