The Belgian international struck as the hosts dominated a one-sided Europa League encounter, leaving Francesco Guidolin's men with a mountain to climb in the return
The result was a suitable reflection of the balance of the game, with the home side consistently threatening the Italians' goal, and eventually got the advantage they deserved.
AZ displayed their intent from the kick-off as full back Simon Poulsen continually looked to provide an outlet for his side as Udinese failed to establish themselves early on.
The first shot of the encounter came from Rasmus Elm, who brought a fine save out of the visitors' custodian Samir Handanovic following a neat interchange with Jozy Altidore.
Antonio Floro Flores looked relatively lively in the early stages but was starved of service and support from his team-mates, who defended resiliently during the opening half.
The home side sustained their pressure on the Zebrette defence, but failed to take advantage of all their possession, creating no real clear-cut opportunities, as Elm and Brett Holman troubling the visitors' defence throughout the half.
Udinese’s first effort on goal came from Pablo Armero, whose deflected effort pressed home keeper Esteban into action. From the resulting corner, the visitors nearly took the lead as Danilo forced an outstanding save from Esteban.
Elm then had a free-kick which was well held by Handanovic as it bobbled through a crowded area of players, missing a deflection which could have posed problems.
As the second half wore on, the visitors began to open up and gradually contribute offensively, with auxiliary forward Armero looking to support Floro Flores in more dangerous areas.
The second half saw Udinese demonstrate some neat passing, sensing an opportunity to take advantage of AZ’s inability to score, but their impressive resolve came unstuck just after the hour mark.
Altidore checked his run superbly to cross in for Martens to plant home a reaction header and send the home supporters into raptures.
The goal helped the tie as Udinese found themselves in a position where they now had to take the game to their opponents. Rather than sitting back, AZ continued to attack and double their advantage.
With the final whistle approaching, both sides went for the pivotal next goal and the home side struck following more fine play from Altidore.
The bustling forward played a smart one-two and sent in a dangerous cross with Falkenburg waiting to complete the simplest of finishes to double his side’s lead.
It could have been even worse for Udinese had Handanovic not produced a fingertip stop to keep Falkenburg out from doubling his personal tally, but the result means Udinese face an uphill task to progress in the competition nonetheless.