Lee Molyneaux’s side dominated proceedings against their BetVictor Southern League Southern Premier South opponents.
But they had to make do with a share of the spoils as they hit the woodwork three times.
Rory Williams was the first to be denied by the goalposts as his deflected effort smacked the upright in 12 minutes.
Andreas Robinson saw his header rebound off the crossbar on 30 minutes, before Matt Paterson’s headed effort had the same end result on the stroke of half-time.
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Gosport continued to dominate after the break.
Theo Lewis’ effort went just wide after the restart, while Ryan Woodford’s header drifted past the post on the hour mark as Borough had to accept it wasn’t to be their day in front of goal.