A goal ten minutes from time by Matthew Dolan gave the visitors victory at a damp CNG Stadium.
After early Boro pressure,Town replied with an excellent move featuring Milne,Wright and Hardy,which ended with a timely tackle from Hines denying Johnny Allen.
The young Middlesbrough side were producing some brilliant football and Chris Elliott in the home goal made two fine saves from Ryan Brobbell to prevent the visitors taking the lead.
However, he was powerless to keep out a thirty second minute 30 yard blockbuster from Ben Gibson.
Harrogate Town were level five minutes later when Boro keeper Luke Coddington brought down Allen in the box and Steve Turnbull made no mistake from the spot.
Middlesbrough upped the pressure after the break and Bruno Pilatos rattled the post with a powerful drive.
A large number of substitutions took place as both clubs sought to give players a game.
Town were denied what would have been an excellent draw when Pilatos broke down the right and his pinpoint centre was headed home by Dolan.