The score was level at 1-1 after 90 minutes.
Lloyd Patterson gave Basford an excellent start, powerfully heading in from a free kick after four minutes.
Zak Goodson then went close, but it was the visitors who were looking increasingly dangerous and Basford were grateful to hear the half time whistle.
The pressure continued in the second half, with York’s Ethan Henderson heading against the bar.
The pressure finally paid off after 72 minutes, Carlie Jebson-King equalising from the penalty spot after a foul.
Goodson was denied by York keeper Ryan Whitley and Basford were agonizingly close to winning the game in added time, Niall Hilton’s shot, which seemed to be drifting wide, hitting the inside of the post.
Straight to penalties it was, Basford scored first, then Dennell Stokes saved York’s first kick, parity was restored as Basford’s second kick was saved and it was 3-3 with one kick each left.
Patterson scored and York’s kick was blazed over, leaving Basford to join South Shields, Chasetown and Maidstone United in the semi-finals.
Updated 09:38 - 21 Feb 2019 by Neil McGowan