The goal came on 54 minutes when Peter Pawlett picked out Niall McGinn on the left, and his low cross was perfect for Zola to steer into the bottom-right corner from close range.
United started the match brightly and John Rankin shot narrowly wide with an effort that skipped off Dons defender Mark Reynolds.
Zola almost took advantage of a poor Gavin Cunning back pass but was thwarted by United keeper Radoslaw Cierzniak, who advanced to block.
Gunning then had a net-bound header blocked by a Dons defender at a corner and Keith Watson was unlucky to see an audacious overhead kick squirm narrowly wide from the rebound.
United were denied an equaliser when Nadir Ciftci and Gary Mackay-Steven combined to put David Goodwillie clean through but Jamie Langfield raced off his line to deny the striker.
Ciftci then clipped the roof of the net with a free-kick soon after but Aberdeen held on to take the three points.