The two teams met at Soulton Road for the Division Two clash.
And it was the visitors who came away with the derby bragging rights mainly thanks to a brilliant undefeated innings from Weston.
The former Cuckfield player faced 157 balls and struck 17 fours as Bridgnorth raced to 246-5 from their 55 overs.
That total made them the only side in the division to reach 200 runs on the day.
In reply, three of Wem’s top four – Harry Chandler, Sam Peate and Chris Peel – were all out for a single. But Mohit Kale (48) and Matthew Cohen (37) steadied the ship to see the score to 92.
Kale was then dismissed by Matty Simmonds, who impressed throughout, taking three wickets for just 33 runs.
Cohen then fell two runs after Kale at 94-5 which soon became 96-6 and then 107-7.
At that point Wem put up the shutters to finish on 142-7.
The result means Bridgnorth sit sixth in the table, one place above Wem in seventh.
Oswestry remain second bottom after they were comfortably beaten on home turf by Himley. The Shropshire side won the toss and elected to bat but were dismissed for 159 runs.
Warrick Fynn was their top scorer with the bat finishing on 65.
Himley were then guided to a five wicket victory thanks to a composed 73 from Callum Lea.
In Division One, Shrewsbury seized an opportunity to lift themselves above Walsall and into ninth place in the table with a resounding 199-run success against Ombersley.
Taking advantage of Ombersley’s decision to field first in perfect batting conditions, Shrewsbury piled up 297-9 before declaring after 53 overs.
Oxford University student George Hargrave led the way with 126, an innings that included 15 fours and three sixes.
Will Parton (69) shared a third wicket partnership of 134 with Hargrave.
Dan Humes (28) also made a solid contribution as Shrewsbury rattled up their highest total of the season.
Liam Gwynne (4-57) and Elliot Mason (4-66) were Ombersley’s best performers with the ball.
But their long afternoon in the sun took its toll as they crashed to 17-5.
Richard Jones hit 29 and tail-enders Matt Davis and Jamie Hammond scored 22 apiece.
But Ombersley were bowled out for 98 in 26.1 overs.
Bradley Thornton took an impressive three wickets for just 19 runs for Shrewsbury.
While Harry Cooke, Luke Thornton, and Lewis Evans all took two wickets apiece.
The match between Walsall and Shifnal was declared void because of Covid issues in the Shifnal camp.