Tottenham dominated possession from the kickoff, but after failing to capitalise on a flurry of early chances, AZ eventually grew into the game and looked to create problems on the counterattack at every opportunity.
The match followed that trend for the rest of the first half, with Spurs attempting to ramp up the pressure just before halftime, but a resolute AZ defence limited them to only long-range efforts, leaving the scores level heading into the break.
Postecoglou's side came flying out of the blocks to start the second half, with Mikey Moore particularly impressing with several weaving runs, and Spurs finally found their opener when Richarlison cheekily chipped in from the penalty spot after Lucas Bergvall was brought down by Maxim Dekker inside the box.
The Lilywhites continued to dominate possession after the goal, but AZ gradually found their rhythm again, threatening Fraser Forster's goal with several flowing counterattacks, but they failed to find the quality in the final moment as they remained behind.
The visitors were reduced to ten men late in the game when David Moller Wolfe received his second yellow card for a late challenge on Brennan Johnson, leaving AZ with even more work to do as they tried to equalise.
Despite AZ attempting to increase the tempo in the final minutes in their last desperate attempt to find a leveller, Spurs were able to hold firm and secure their 1-0 win.
TOTTENHAM HOTSPUR VS. AZ ALKMAAR HIGHLIGHTS
Richarlison goal vs. AZ Alkmaar (53rd min, Tottenham Hotspur 1-0 AZ Alkmaar)
Cheeky from Richarlison!
Spurs are rewarded for their flying start to the second half as Lucas Bergvall is brought down inside the box with the referee signalling for a penalty, and after a brief discussion between James Maddison and Richarlison, the Brazilian eventually takes charge of the spot kick.
The striker steps up, stutters his run, and calmly lifts the ball into the centre of the net as the goalkeeper dives out of the way, giving Spurs their deserved lead and scoring his first of the season.
85th min: David Moller Wolfe (AZ Alkmaar) red card
Wolfe receives his second yellow card after bringing down Brennan Johnson with a poorly executed challenge, the defender's second red card in as many matches for AZ!
MAN OF THE MATCH - DESTINY UDOGIE
Udogie was superb throughout the 90 minutes in both defence and attack for Spurs tonight, making two interceptions, two tackles and blocking one shot to help the hosts retain their cleansheet.
The defender also had 74 touches in the 90 minutes, making 47 passes, one of which was a key pass, and was a tidy operator as Tottenham controlled the possession of the game.
Moore must also be mentioned, despite admittedly enduring a quiet first 45 minutes, he came to life after being switched to the left hand side at the beginning of the second half, causing plenty of problems for the AZ defence before eventually coming off.
TOTTENHAM HOTSPUR VS. AZ ALKMAAR MATCH STATS
Possession: Tottenham Hotspur 59%-41% AZ Alkmaar
Shots: Tottenham Hotspur 16-9 AZ Alkmaar
Shots on target: Tottenham Hotspur 5-3 AZ Alkmaar
Corners: Tottenham Hotspur 6-4 AZ Alkmaar
Fouls: Tottenham Hotspur 13-17 AZ Alkmaar
BEST STATS
WHAT NEXT?
Tonight's win is Tottenham's second consecutive victory and their seventh in the last eight matches, and they will aim to extend that superb run when they travel to face Crystal Palace in the Premier League on Sunday.
AZ's fifth consecutive defeat marks a stark downturn from their seven match unbeaten run across all competitions at the start of the season, and they will look to end that losing streak when they host Go Ahead Eagles in the Eredivisie on Sunday.