Bryan Morris had already hit the bar by the time a dominant Boro edged ahead after 12 minutes. Junior Mondal was fouled in the penalty area and Callum Cooke drilled home powerfully from the spot.
Cooke and Boro enjoyed a very productive first half and just eight minutes and another strike on the woodwork later, this time from Callum Johnson, a second was added when Robbie Tinkler rose to head home a deep cross into the box.
On the stroke of half-time Junior Mondal made it three when he tapped in at the back post to round off another cutting move.
By then Thackeray in the Northallerton goal had made an outstanding save, but there was little he could do as Boro added a fourth on 52 minutes when Joe Wheatley, one of two changes at half-time, tucked home neatly at the back post.
This was a very competent display from Paul Jenkins' side who had their attitude and approach right from the start, ensuring the holders progressed.
Fryer; Johnson, McGhee, Elsdon, McGoldrick, Kitching (Coulson 62), Morris (Wheatley 46), Tinkler, Mondal, Cooke (Capt.) (Pattison 46), McAloon. Sub. Hemming (gk)