Broadstone, who did the double over Poole last season, dismissed their rivals for 148 with Trembath (4-13) and Abbas (5-41) putting them to the sword at Poole Park.
Although Poole had made an encouraging start thanks to Nick Evans (53) and Ciaran Quinn (32), they collapsed from 106 for two as Trembath and Abbas made hay.
Poole skipper Kristian Wilson (3-21) removed openers Luke Carter (25) and Pete Russell but, with Sean Hatton (26) chipping in, Broadstone kept up the momentum.
The chase was far from convincing and James Rimmer (17) and Trembath (15) helped to see Broadstone over the line at 152 for seven, with no fewer than 32 wides adding their cause.
Poole captain Wilson said: “We had the game nicely set up and Broadstone had gone quiet. But a middle-order collapse meant we failed to use all our overs. It was a great effort in the field but we left ourselves too much to do.”
Wimborne & Colehill avenged two defeats by neighbours Ferndown Wayfarers last season by recording a comprehensive 95-run triumph over their rivals at Dolman's Farm.
An opening stand of 131 between Colin Randall (105) and Tom Caines (43) laid the foundations as Wimborne & Colehill posted a formidable 237 for five.
Randall also shared 59 for the second wicket with Joe Weld (34) - with Daryl Baines (2-37) removing the pair - before Simon Woodruff (26) boosted their total.
The Wayfarers reply started disastrously and, after Chris Cornish (2-31) had bowled Lee Charlton, Gavin Will (3-14) helped reduce them to 27 for four.
Ben Ratcliffe (2-31) and James Williams (2-14) were also among the wickets, while Ferndown's 142 owed much to a defiant last stand of 40 between Niall Snook (31*) and Andre Jensen (23). Alex Sansom (26) also chipped in for Wayfarers with Wimborne & Colehill sending down 32 wides.
Champions Parley bounced back following their shock defeat by Swanage and dished out an eight-wicket hammering to Stalbridge at Dean Park.
Ollie Green (3-31) and one wicket apiece for Jamie Hicks, Luke Condron, George Wilson and Ollie Shrubsole restricted the visitors to 187 for seven.
Tom Jacques (56), together with unbeaten knocks from Adam Cocking (42) and Green (51), saw Parley coast to victory at 188 for two from just 25 overs.
Swanage came down to earth with a bump following their heroics against Parley and were condemned to a three-wicket defeat by Weymouth.
Ricky King (21) and Jem Warner (33*) helped Swanage post a modest 119 for nine before King and Elliott Baczala both took three wickets as Weymouth edged home at 122 for seven.