Town’s skipper Tim Fordon won the toss and put the visitors, title challengers, Hull Ionians Coal Exporters into bat.
Hull Ionians Coal Exporters posted a big score with Will Parker (92) and Paul Broddy (83) playing key roles in them closing their innings on 309/7.
While Hull did bat well Graham proved to be a the pick of the bowlers for Beverley as he finished with a credible four wickets for just 61 runs.
In reply Town kept their cool and while they finished some distance off the total Hull Ionians Coal Exporters were unable to bowl them out to claim the outright win.
While the result has little impact on Town the result does see a change at the top of the Y&DSCL Division Two Holderness with Pocklington knocking Hull off the top.
Fordon will be looking for his side to dig in deep again next weekend as Town will for the second consecutive week take on the league leaders as Pocklington visit Norwood.
When the side met in May Pocklington cruised to a 110 run win, a result they will be looking to emulate to keep their championship aspirations alive.