Centre Natasha Greville defended well and along with winger Felicity Powdrell made the most of their opportunities on attack. Iosefo, Greville and Cottrell scored the Tuis tries and Cottrell kicked two conversions and two penalties.
The Hawke's Bay Saracens opened their Hurricanes competition with a 96-15 thrashing of Wellington Maori at Park Island on Saturday.
Magpies winger Michael Buckley and loosie Fa'alemiga Selesele, who took the field as a substitute, both scored hat-tricks. Loosie Matt Gardner also shone with his impact off the subs bench.
The Hawke's Bay under-19s secured a top eight berth at their national tournament which begins in Taupo on Saturday with the weekend's 55-24 win against the Hurricanes Heartland under-20s at Park Island.
"We trailed 12-8 at halftime and 19-8 early in the second half and then found our groove," Bay manager Mike Smith said.
Magpies squad members Lincoln McClutchie and Danny Toala both displayed their class at first five-eighth and fullback respectively.
Second five-eighth Clerk Kopelani also impressed in the Bay backline.
Flanker Trei Nepe-Apatu and prop Lee Moleli both had high workrates in the Bay pack.