Swansea failed to beat Conference-bound York, leaving boss Kenny Jackett to wait for his first home win.
O'Leary forced a save deep in the second half
York keeper Chris Porter saved well from Paul Connor and Kris O'Leary as the Swans got the better of the play.
Connor found the net midway through the first-half, but his effort was ruled out for offside.
Roger Freestone in the home goal only had to deal with one shot on target all game as York were confirmed as the bottom club in the league.
Not even the introduction of top-scorer Lee Trundle as a substitute could inspire the Swans.
Connor again went close in the second half but it was a low-key ending to the
Swansea: Freestone, Stuart Jones, Tate, Iriekpen, O'Leary,
Coates, Maylett, Martinez, Britton, Nugent, Connor. Subs:
Trundle, Roberts, Corbisierso, Rees, Murphy.
York: Porter, Law, Merris, Brass, Hope, Wood, Dunning, Stewart,
Arthur, Ashcroft, Yalcin. Subs: George, Stockdale, Staley,
Referee: S Tanner