Robinson, who was relegated to the substitute bench in favour of his uncle John Ryan, struck on 46 minutes, tapping home after Payne fumbled the ball from a Jonathan Prizeman shot.
Keedy and Prizeman both had opportunites to extend the lead for the home side as the Lily Whites got their season back on track after the midweek 2-0 defeat to St Patrick's Athletic.
City almost grabbed a point with just minutes left in the clash, after substitute Jamie Hughes was taken down in the box, but Dundalk keeper, John Connolly saved to earn Martin Murray's men all three points.