Playing with the wind advantage in the first half, Lealholm took the lead after five minutes.
Good work from Thompson led to an excellent cross which found Kev Brown who scored from close range.
Brown was again the provider when he put through Morgan who pushed the ball past the keeper to extend Lealholm's lead.
Lealholm remained comfortable for most of the half although the visitors began to pressurise towards the end and were unlucky not to pull a goal back.
Five minutes into the second half, a good team move led to Thompson finishing well to give Lealholm a third.
Newlands Park then took control and 10 minutes later, they produced two goals in two minutes.
The first was a headed goal inside the area then a shot from 30 yards was carried by the wind into the net.
Lealholm then had their backs to the wall as the visitors looked for the equaliser but the Lealholm defence, with captain Knight at the helm, held firm as they hung on for a place in the next round.
Result: Newlands Park Hotel 2 Lealholm 3