On a night billed as win-or-bust for Postecoglou and his side, they overcame a 1-0 deficit from the first leg last week to win 3-1, with a James Maddison strike in between Odobert's first two goals for Tottenham.
The 20-year-old French winger, signed from Burnley last August, has missed most of the season with a hamstring injury but was back with a bang to see Spurs through to a last eight showdown with Eintracht Frankfurt.
It was far from comfortable, though. Tottenham had not troubled AZ's goalkeeper Rome-Jayden Owusu-Oduro until Odobert scored his first in the 26th minute. The goal was an error from the Dutch team, with a loose back pass putting Wouter Goes under pressure in the right-back position.
Nevertheless it was Heung Min Son's determination to close down the defender that forced a rebound into the path of Dominic Solanke, who was loitering in the penalty area with intent. The Spurs striker could have tried a tricky shot but smartly chose to lay the ball square across goal, where Odobert met it with a left-footed shot into the roof of the net from 12 yards.
Spurs were level on aggregate and fired up. Son looked like he wanted to prove a point with a couple of driving runs into the penalty area, the first ending with Lucas Bergvall curling a shot over the bar, and the second with Odobert slicing the ball high into the air.
From the start, the Dutch side seemed content to try containment rather than go looking for another goal to build on their 1-0 lead. Ernest Poku, so dangerous in the first leg, had precious few opportunities to show his pace or trickery, and when he had a half-chance, Guglielmo Vicario smothered the ball at his feet.
Troy Parrott, AZ's top scorer with 17 goals before kickoff, got little change out of the returning Micky Van de Ven and Cristian Romero, who caught the Dubliner with a thunderous challenge early on.
And within three minutes of the restart, James Maddison put Spurs ahead in the tie for the first time in the tie. Pedro Porro won the ball in the Tottenham half and sent Odobert away. The winger played the ball forward to Maddison, who exchanged passes with Son before steering a precise shot past Owusu-Oduro.
Finally AZ started to attack in earnest. Poku got clear of the last line of defence, but Van de Ven used his astonishing spped to make a recovery tackle. When the big Dutchman went off to be replaced by Archie Gray on the hour mark, he received a standing ovation from home supporters.
But within minutes AZ levelled the tie after a mix-up in the Tottenham defence. Odobert did well to intercept a pass towards Peer Koopmeiners, but a lack of communication with Bergvall allowed the Dutch midfielder the chance to shoot, which he did with power past Vicario.
The scores were level on aggregate, the game was a knife edge, and nerves kicked in.
Then Odobert struck again. It was a flowing move from one goalmouth to the other, perfectly illustrating 'Angeball' at its best. Vicario, Gray and Romero played short passes out of defence to Maddison, who drove forward through midfield. The ball was shuttled down the left via Son to Spence, who drilled in a low cross that was flicked on by Solanke to Odobert, who converted from close range at the far post.
Still there were chances for AZ to equalise. Kees Smit went on to add height to their forward line, but headed over from close range when Parrott looked to be in a better position to score. Smit then upset his team-mates when he volleyed wastefully wide in the closing stages.
The final whistle was greeted with relief as much as celebration, as Tottenham's fans realised their season is still very much alive.
Tottenham (4-3-3): Vicario 7; Porro 7, Romero 6, Van de Ven 8 (Gray 61), Spence 8; Sarr 6, Maddison 7 (Bissouma 77), Bergvall 7 (Davies84); Odobert 8 (Johnson 78), Solanke 6, Son 7.
AZ (4-3-3): Owusu-Oduro 6; Maikuma 6 (Meerdink 84), Goes 5 (Belic 89), Penetra 6, Moller Wolfe 6; Clasie 6, Buurmeister 6 (Smit 65), Koopmeiners 7; Sadiq 5 (de Wit 46), Parrott 6, Poku 6.