Wellington midfielder Nati Hailemariam burst through the lines in the 40th minute and slipped a pass to Bouzoukis but his tame shot was saved by Rovers keeper Oscar Mason.
The visitors took the lead a few minutes later when a corner kick played over to the back post was squeezed home by defender Ben Mata.
Napier had a decent chance to level the scores before halftime, getting numbers forward on the counterattack, but Cameron Emerson's touch inside the box was heavy and it remained 1-0 at the break.
Former Hawke's Bay United forward Jesse Randall hit the post from a tight angle a few minutes after halftime.
Napier City Rovers forward Leaford Allen could only scuff his 52nd minute shot on goal after a good flick on from Emerson, creating an easy save for Olympic keeper Scott Basalaj.
The visitors hit the post again through Bouzoukis, but he made no mistake five minutes later to make it 2-0 on the hour mark.
Former Rovers forward Kailan Gould grabbed two goals in the final 20 minutes; both tap-ins after a cut back from the byline.
Between those goals Jonny McNamara missed a penalty for Napier City Rovers on a day to forget for the hosts.
Olympic finished the game with 10 men after defender Justin Gulley received his second yellow card for a rash challenge in added time.
Napier City Rovers won't have to wait long for a shot at revenge – their next home game on July 24 sees Wellington Olympic back in town for the Central League reverse fixture.
Rovers are away to Petone next weekend, while Havelock North Wanderers are home at the Hawke's Bay Regional Sports Park for the visit of Miramar Rangers on Sunday afternoon.
