Central then took control with the ball to remove Indian Sports for 118, to claim a 141-run victory.
Andrew Gibbs took the key wicket of Northern Knight Bharat Popli, while Mark Chapman, Trembath and Graeme Whiles all took two wickets apiece.
Trembath said it was nice to notch up their first win of the season.
"The boys really wanted to bat well on Saturday and I think the 115-run partnership for the first wicket might have been a Central record."
He said it was Barrow's first 50 in premier cricket and he batted very well.
Trembath said hopefully they could build on that performance during the coming weeks.
The new wicket block at the centre of Smallbone Park, which was recently replaced for the first time in over 20 years, was used for the first time during the derby match on Saturday.
Trembath said it played really true for a new wicket.
They expected it to be a bit slower and softer early on, but it played the exact opposite.
"It played bloody well at the start and got a bit slower towards the end of the day."
He said it was a nice upgrade to Smallbone - the home of Rotorua cricket.
The Baywide Cup is the first of three competitions played during the Baywide premier cricket season, followed by the T20 competition and all-important Williams Cup. The Baywide Cup is run as a 106-over match. The Baywide Cup takes a break next weekend for the opening two rounds of the Lonestar T20 competition.
- Baywide Cup results (round four):
Otumoetai Cadets 103 (Joe Carter 68; Campbell Thomas 4-9, Tommy Clout 3-29) lost to Greerton 104/3 (Tom MacRury 53no; Sean Davey 2/19)
Mount Maunganui 121 (Mark Divehall 38, Tony Goodin 25; Brody Gilroy 5-43, Alex Gooding 3-42) lost to Te Puke 122/8 (A Gooding 61, Stephen Crossan 34; Luke Tatley 3/19)
Bayleys Central 259 (Johnny Barrow 78, Jason Trembath 56; Yohann Irani 3-12) defeated Bay of Plenty Indian Sports 118 (Magan Vashisht 27; Mark Chapman 2-14, Jason Trembath 2-16, Graeme Whiles 2-22)
Tauranga Boys' College the bye.
- Points: Te Puke 86, Cadets 66, Greerton 66, Tauranga Boys' College 26, Central 23, BOP Indian Sports 3, Mount Maunganui 3.