Neither side were brilliant or particularly adventurous in a dull first half, but it was Tottenham who got the breakthrough when Brennan Johnson bundled in from close range.
After that it was all about defending what they had.
In the second half they had just one touch inside the opposition box, did not have a shot on or off target, and just 19.8% possession.
And in uncharacteristic fashion Postecoglou sacrificed an attacking player for a defensive one when Johnson was replaced by centre-back Kevin Danso.
Former Tottenham goalkeeper Paul Robinson said: “It was a poor watch, it really was an awful game, but Tottenham ruined it. They spoiled the game. That was the game plan.
“It was not what we are used to seeing from Ange Postecoglou. It was a gritty, horrible, defensive, determined, rugged performance.”
Tottenham certainly rode their luck at times and also needed some big individual performances as Micky van de Ven made an incredible goalline clearance from Rasmus Hojlund, while goalkeeper Guglielmo Vicario made a big save from Luke Shaw late on.
Overall Spurs had three shots, just 27.7% possession and completed 115 passes, all the lowest figures Opta has on record by any team in a major European final (since 2009-10).
“This showed there is a plan B,” former Tottenham forward Gareth Bale said on TNT Sports.
“I guess as a manager if Ange Postecoglou also matures, and brings that type of defensive side to his teams, of course he can still have great success here.”
Ex-Leicester midfielder Robbie Savage added on BBC Radio 5 Live: “He’s adapted, big Ange.
“He hasn’t gone a full press, he hasn’t gone with a high line. He sat deep and won the game because he adapted.”
