Electing to bat on Wednesday in the first match of the two-Test series, Sri Lanka slumped to 4-106 but Kamindu's fourth Test hundred ensured the hosts breached the 300-mark.
Ramesh Mendis was batting on 14 at stumps with Prabath Jayasuriya yet to open his account at the other end at the Galle International Stadium.
Earlier, New Zealand speedster William O'Rourke (3-54) hectored the Sri Lankan top order bowling with unbridled aggression.
Dimuth Karunaratne (two) was done in by the extra bounce O'Rourke generated while Pathum Nissanka (27) succumbed to a searing yorker.
Angelo Mathews (36) combined with Dinesh Chandimal (30) to arrest the slide but had to leave the field after getting his right index finger bruised by an O'Rourke delivery.
New Zealand captain Tim Southee removed Chandimal, while his Sri Lanka counterpart Dhananjaya de Silva (11) fell to Glenn Phillips, who struck with his second delivery.
Mathews returned only to be dismissed by O'Rourke at the stroke of tea.
Kamindu went into the match with seven 50-plus scores in 10 Test innings and got a reprieve on 21 when Phillips spilled a catch at midwicket.
The left-hander drove spinner Rachin Ravindra through cover to bring up his hundred, which earned the 25-year-old a standing ovation from his teammates in the dressing room.
He raised 103 runs with Kusal Mendis, who made 50 before his bizarre dismissal in the final session.
Kusal pulled a half-tracker from Phillips straight into the body of Tom Latham at short-leg. The ball hit Latham's arm and lobbed up for Southee, stationed at midwicket, to dive forward and take the catch.
Ajaz Patel ended Kamindu's stellar knock when the ball landed on footmarks and reared.
Kamindu, who had shaped for the cut shot, could only glove it to Daryl Mitchell, who moved across from slip to take the catch.
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