As a result, they will now travel to face York City in the fifth round after the draw was made on Monday.
The game will be played on Saturday, February 12.
Against Boston, a brace from Jack Foalle as well as a stunning strike from Liam Henderson and a Liam Noble penalty helped the Highwaymen progress despite strikes from Jake Wright Jnr, Scott Duxbury and Danny Elliott had given the Pilgrims hope.
In what was a cracker of a cup tie, Morpeth trailed 0-1, led 2-1, went 2-3 down with two goals in first half injury time, levelled at 3-3 with a Noble penalty before Foalle, with his second of the game, snatched the winner in the 70th minute.
He had a chance to complete his hat-trick ten minutes later when he was put clean through on goal, but unfortunately lifted the ball over the top.
The game was watched by a bumper crowd at Craik Park, including Prof Sir Jonathan Van Tam, England’s Deputy Chief Medical Officer, who is a staunch Boston supporter.
After the match manager Craig Lynch said: “For me there was only one team that looked like winning it. The lads are buying in to what we are wanting to do here.
"We’re into the last 16 now and it’s still a long way from Wembley but we can still dream about it.”
Two goal hero Foalle said: “It’s great. I just constantly want to keep getting into the next round. With the belief in the management team behind me it has given me the motivation to go out and stick the ball in the back of the net.”
Morpeth, who have now climbed to 12th in the Northern Premier table following a good run of results since Lynch’s appointment, are set to return to league action this Saturday (January 22) with a home game against Buxton, who are currently sitting fourth.
Then, next Tuesday (January 25), they are back at Craik Park for another league match against 15th placed Stalybridge Celtic.
“It’s all about one game at a time,” said Lynch, “but the lads are on a great run of form at the moment.”