Gainsborough: Beesely is the hat-trick hero as Trinity beat Boston in the cup
Beesley gave Trinity a first-half lead but they were pegged back after the break when Carl Piergianni equalised for the hosts.
However, Trinity – and Beesley – were not to be denied and two further goals gave the striker his treble and Trinity the win.
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Hide AdBoston, meanwhile, finished with 10 men after Zak Mills was sent off late on for a second bookable offence.
Gainsborough will now face Conference Premier side Grimsby Town in the semi-final at The Northolme on Saturday (2nd August, kick-off 3pm).
Meanwhile, striker Nathan Plumtree and midfielder Charlie Binns have both signed for Trinity on non-contract terms.