Number eight Kaino, a key member of Steve Hansen's New Zealand squad, came off the pitch after 18 minutes with a damaged finger, while outside back Piutau suffered what the Blues suspect was ligament damage to his knee.
Blues winger Melani Nanai scored his first two Super Rugby tries, Kaino's back row replacement Steven Luatua also crossed and teenager Akira Ioane secured the bonus point with a 50-metre run to cap another outstanding display on the blindside flank.
Replacement flyhalf Ihaia West scored the home team's fifth try to ensure just a second victory of another disappointing season for John Kirwan's side.
"We wanted to be physical and be really direct with them," Blues co-captain Keven Mealamu said in a pitchside interview at Eden Park.
"It was nice to get a good result for our people."
The Force were already 31 points down when prop Pek Cowan got them on the scoreboard with a 51st minute try but further scores from flanker Matt Hodgson, scrumhalf Ryan Louwrens and number eight Ben McCalman secured them a bonus point.
That will be scant consolation for the Perth-based outfit, who have now lost 10 straight matches since upsetting the reigning champion New South Wales Waratahs in their season opener.
"Our attack is going well but our defence has got a lot to make up," Force skipper Hodgson said.
"You can't let a team get away with that many tries early on in the game (but) we'll keep fighting to the end of this competition."