The early minutes in the DSB Stadion saw little goalmouth action at either end, although AZ enjoyed the better of the possession as they sought to overturn a 4-2 first-leg deficit.
Fourteen minutes in Shay Given did well to tip away Julian Jenner's flick from Shota Arveladze's excellent ball in from the left flank, but within seconds the home side were ahead.
Danny Koevermans found time and space on the right and his low cross to the near post was tucked home by the unmarked Arveladze from six yards out.
Magpies skipper Scott Parker fired an effort narrowly over the crossbar soon after, before the game quietened back down into its earlier pattern.
Ten minutes before the break Given was out quickly to gather after Jenner threatened inside the area and shortly before half-time, Paul Huntington was lucky not to concede a penalty as he brought down Jenner.
Nine minutes after the restart a quickly-taken Nolberto Solano free-kick released Kieron Dyer, but he prodded his shot agonisingly past the far post after rounding keeper Boy Waterman.
Two minutes later AZ were ahead in the tie as Maarten Martens swung in a left-foot corner from the right and Koevermans rose highest at the near post to head home.
Given had to push away a Jenner effort and Steven Taylor recovered from his own error to clear Koevermans' shot off the line as AZ took total control.
Huntington nodded a Moussa Dembele drive off the line, before Obafemi Martins wasted a clear chance to rescue the tie, failing to beat Waterman after being played in on the left by Dyer.
Parker then had a goal disallowed for some pushing, before Given made a superb save from close-range following Arveladze's turn and shot.
However, AZ had done enough to book their place in the last eight and leave the Magpies' European dreams in tatters.