The Minstermen were unable to add to the advantage given them by Jamie Reed from the penalty spot on 14 minutes as the visitors defended valiantly against a tidal wave of second half pressure.
After Jake Howells was sent off for the handball that gave City their penalty, team-mate Keith Keane joined him for an early bath when he was dismissed on 54 minutes for a second bookable offence.
But despite laying siege to the visitors’ goal, York will go into next Saturday’s re-match at Kenilworth Road with the slenderest of leads as they seek to book their place in the final at Wembley.
Reed looked the likeliest to increase City’s advantage in the second period.
He had a 25-yard free-kick brilliantly tipped over the bar by Mark Tyler in the 53rd minute and came close again with a 10-yard effort.
York regularly tested Tyler from distance and moved the ball sweetly in the afternoon sunshine.
Luton were practically anonymous as an attacking threat in the second half but the tie remains intriguingly balanced.
York City: Ingham, Oyebanjo, Meredith, Smith, Parslow, Reed, Kerr, Fyfield, Challinor, Blinkhorn, Ashikodi. Substitutes: Chambers, Pilkington, Potts, Blair, McLaughlin.
Luton Town: Tyler, Taylor, Keane, Pilkington, Lawless, Crow, Howells, Asafu-Adjaye, Henry, Watkins, Kovacs. Substitutes: Blackett, Osano, O’Donnell, Kidd, Woolley.
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