With their first chance of the game Morpeth went ahead with Liam Henderson seeing his initial effort saved before netting the rebound on 15 minutes.
The Boro came back into the tie during the first half and Daniel Burns rifled his effort over the bar following Jonathon Moran’s header down.
Bradley Grayson then had two chances for Rangers misconnecting with his first shot before racing away from the defence and blazing over.
With three minutes before the break Grayson was again through on goal twice before the break wasting the first effort and then denied by Dan Lowson at the second attempt.
Any hopes of a second half revival were dented in the 56th minute when Chris Reid powered home a header.
Burns was given a straight red card three minutes later for an off-the-ball incident with Carl Finnigan.
Grayson almost pulled a goal back but was again denied by Lowson.