The Toffees had already ensured their passage to the knockout stages by winning their first three games and boss David Moyes took the opportunity to rest several key players for the trip to the Netherlands.
The Toffees made a perfect start in Alkmaar as Andy Johnson broke the deadlock with little over a minute on the clock.
Following a throw-in, the England striker outmuscled his marker on the left-hand side of the penalty area before driving a low, right-foot shot past goalkeeper Boy Waterman at his near post from 15 yards out.
With 16 minutes on the clock it was 1-1 as Gretar Steinsson fed Graziano Pelle on the edge of the Everton area and he fended off Phil Jagielka to drill a right-foot shot beyond the despairing dive of Stefan Wessels.
However, the Toffees were back in front shortly before half-time as Johnson's ball in following a corner was thumped goalwards by Joleon Lescott and the rebound popped up for Jagielka to head home from close range and record his first goal for Everton.
Moyes' side now looked to be in control of the game, but midway through the second half they conceded a second equaliser, as some slack marking from Julian Jenner's left-wing corner allowed Kew Jaliens to power a header home at the far post.
Everton were not to be denied, however, as they brought an end to AZ's impressive unbeaten record in European competition.
Eleven minutes from time Victor Anichebe's through ball gave James Vaughan the chance and he sidefooted past the advancing Waterman from an acute angle to secure the win.
Anichebe almost added a fourth himself in stoppage time, but Everton had done more than enough to earn a fifth win in a row in all competitions.