A chip-and-chase try from former sevens star Kurt Baker, back in Manawatu after a stint in Italy, extended the home team's lead, with a 40th minute Black penalty capping off a solid first half.
Tasman, beaten 39-0 by Canterbury last week, scored their first try of the season five minutes after the restart. Skipper Alex Ainley ran onto Jordan Taufua's well-timed off-load, Hunt made the conversion and Tasman were back in the game at 15-10 down.
Hooker Hunter Prescott restored the Turbos' margin, when he barrelled over in the midst of a well-executed lineout drive, only for Tasman to narrow the gap with another Hunt penalty.
They closed to within two, when Michael Green finished off Will Jordan's smooth break with 15 minutes remaining, but had to play most of the final quarter with 14 men, first losing wing Tima Faingaanuku, then Taufua to the sin bin.
Black gratefully nailed two more penalties, then Hamish Northcott wrapped up the win with his 80th minute try.