Stevenage moved back to the top of the Blue Square Premier with a club-record seventh straight clean sheet.
Morison was on target with Stevenage's first goal
Richard Brodie should have put York ahead when he raced clear but he fluffed his shot to let Borough off.
Steve Morison broke the deadlock after half time when he lashed home a John Martin cross.
On-loan Nick Hegarty saw his volley tipped around the post for York but when Daryl McMahon turned in Adam Miller's pass the points were safe.
York: Henderson, Craddock, McGurk (Parslow 85), Kelly, Robinson, Wroe, Panther, Woolford, Hegarty, Brodie (Sodje 61), Brayson.
Subs Not Used: Evans, Farrell, Beardsley.
Stevenage: Julian, Fuller, Henry, Arber, Nutter (Gaia 14), Dobson (John 63), Lewis, Martin, McMahon (Allen 84), Miller, Morison.
Subs Not Used: Russell, Wilson.
Goals: Morison 46, McMahon 81.
Ref: R Fletcher (Derbyshire).