City's best chance came when substitute Stuart Elliott - on for the injured Dean Keates - crossed to Gary Alexander at the front post, but he was unable to hit the target from close range.
The visitors gave City a warning after 38 minutes when Craig Liddle headed against the woodwork from a Hodgson free kick, with Paul Musselwhite beaten.
They then made the breakthrough in stoppage time. Hodgson sprinted down the left wing, crossed to Conlon who was unable to get a shot in, but Simon Betts at the back post fired past Musselwhite.
City looked brightest after half time, however. Stuart Elliott forced a save from Mike Ingham in the first minute, before Darlington were down to ten men. Phil Jevons was played through on goal, and Andy Liddle caught him from behind. The referee showed him the red card despite strong protests, but City couldn't take advantage of having an extra man.
Stuart Green had the best chance of the game when a tight offside decision went his way. He raced through on goal, but shot straight at Ingham when he really should have scored, or played it square to the unmarked Stuart Elliott who would have had an open goal.
John Anderson was also guilty of an appalling miss soon later. Ingham blocked Green's shot, and then Anderson missed the open goal from close range.
Ingham's crossbar was rattled when Green's cross was nodded onto the top of the woodwork, with ten men Darlington clinging on to their 1-0 lead.
Stuart Elliott had a few more shots on goal but Ingham was not to be beaten, and the curtain came down on Boothferry Park in a frustrating and disappointing manner, with Darlington going home with the three points.
City line-up (4-4-2): Musselwhite; Regan, Joseph, Anderson (Webb 73), Smith; Green, Melton, Ashbee, Keates (Elliott 23); Alexander (Whittle 45), Jevons.
Un-used subs: Deeney, Holt.