United had nine players with senior appearances to their names this season, and that experience was far more superior than was on offer to City's squad.
But City's defence held firm under pressure and initially restricted the visitors to just a few speculative shots.
In fact, the best save of the opening 30 minutes was produced by Whites 'keeper David Lucas, who parried a shot from Adam Smith, but Leeds were unfortunate not to open the scoring just before the interval.
Liam Dickinson was denied when Josh Mimms beat away his volley, and the striker's effort on the rebound was turned against a post by defender Luke Woodall.
Leeds finally broke through York's defence four minutes after the interval when Robinson burst through City's backline and calmly slotted a low shot beyond Mimms.
York City Res: Mimms, Radcliffe, Brown C, Hogg (Knowlson h/t), Woodall, Lisles, Russell, Smith C, Smith A (Emmerson 47), Torpey (Parker 64), Shepherd.
Subs (not used): Qualter, Burns.
Leeds United Res: Lucas, Richardson, Milne, Lees, White, Prutton (Hatfield 83), Johnson, Hughes, Robinson (Falkingham 70), Grella, Dickinson (Elliott 70).
Subs (not used): Edwards, Darville.