Ben Miller, who started for Taieri at No 10 after Josh Casey pulled up lame in the warm-up, scored after just 90 seconds and the Eels never looked back.
Miller played well running the cutter, as Taieri grabbed four more tries to lead 31-5 at halftime.
Taieri prop Nick McLennan scored two tries, his first coming after a bulldozing run from 25m out when he fended off a few would-be-tacklers.
He also dominated at scrum time and consistently put his side on the front foot with ball in hand.
Flanker Vincent O'Connell also stood out for Taieri, making a bunch of good tackles and popping up all over the show to pounce on loose ball.
Green Island was guilty of too often kicking the ball back to Taieri's dangerous back three, and of falling off too many tackles.
It did enjoy patches of dominance, particularly late in the first half, but it struggled to finish.
Green Island prop Ricky Campbell scored a well-deserved double to match McLennan's effort.
