But the Highwaymen had to do it the hard way, coming from 2-0 down after conceding two early goals.
With temperatures in the mid-30s, Morpeth had to endure a torrid opening ten minutes.
The home side opened the scoring after only three minutes before doubling their advantage on nine.
But straight from the restart Town pulled a goal back with a Michael Turner header.
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After an enforcd water break on 20 minutes, Morpeth then drew level after the resumption with Jack Foalle finishing with aplomb after a long through ball from Liam Noble.
The winning goal came in the second half when Connor Thompson cut in from the left and curled a shot into the far corner.
After the game, manager Craig Lynch said: “I thought we managed the game brilliantly.
"We have players with plenty of experience and we want them to use that experience to see us through games.
"We have also got a few young players, but they didn’t let it phase them and ultimately it was a fantastic performance.”
Connor Thompson, who scored the winning goal added: “We had a slow start, but after going 2-0 down we kicked on and looked to be in control. It was a good goal and hopefully it will help me kick on for the rest of the season.”
Morpeth are next in action on Saturday (July 23) away to Dunston.