The visitors extended the lead six minutes before the break when Brad Shields took a quick tap penalty and recycled the ball to Isaia Walker-Leawere who forced his way over to make it 12-0 halftime lead.
The Steamers finally got on the board with a Mike Delany penalty eight minutes into the second half but Wellington continued to control the game when Ben Lam chased down a Jackson Garden-Bachop chip into the corner for a try.
Bay of Plenty offered some slim hopes of a comeback when Terrence Hepetema scored in the 63rd minute with a Delany conversion closing the gap to 17-10.
However Wellington dominated the closing stages of the match with Wes Goosen and Aumua both scoring to guarantee a bonus point victory.
Wellington will look for four straight wins when they host Hawke's Bay on Wednesday while the Steamers look to bounce back in Palmerston North next Friday.