The Highwaymen had never advanced beyond the third round in their 128-year history, but are now just three victories from securing a trip to Wembley Stadium.
They are the only members of the Pitching In Northern Premier League left standing in the competition.
In October, the two sides met in the fourth qualifying round of the FA Cup, with York running out 3-1 winners.
Morpeth reached the fifth round of the FA Trophy last weekend thanks to a thrilling 4-3 victory over National League North side Boston United at Craik Park.
Trailing 3-2 at half-time, they turned things around after the break, courtesy of Liam Noble’s penalty and a second of the game for Jack Foalle.
Speaking after their win in the last round, manager Craig Lynch said: “There are not enough good words to describe that [performance] today; absolutely outstanding.
“I don’t think anyone could say that we weren’t the better team.
“This is what it’s all about. We are still a long way from Wembley, but we’re still in the hat and we can still dream about it.”
All last 16 ties are due to take place on Saturday 12 February.
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