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The Hoops started the second leg at the AFAS Stadion with a 2-0 lead from the first leg and edged through at the expense of its Eredivisie opponent.
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Kyogo Furuhashi continued a great start to his Celtic career by opening the scoring with his seventh goal for the Glasgow club, but Zakaria Aboukhlal gave AZ hope when he equalised only six minutes in.
Starfelt found the back of his own net with 26 minutes gone and AZ ramped up the pressure after the break, but Ange Postecoglou's side clung on to advance 3-2 on aggregate ahead of the Old Firm derby at Ibrox on Sunday.
Furuhashi fired home with his left foot in the third minute, only for Aboukhlal to level with a strike from the centre of the penalty area to punish some poor defending and uncertain goalkeeping from Joe Hart.
Celtic was not doing itself any favours and Starfelt embarrassingly turned the ball into his own net when attempting to clear the ball.
AZ had their chances to force extra time but were made to pay for some poor finishing, with Ernest Poku missing a golden opportunity from close range and Dani de Wit putting a free header wide.
Celtic's six-game winning run came to an end but it held on to qualification as AZ were consigned to the Europa Conference League group stage instead.