Palace looked to have a chance to add a third on the break in with 20 minutes remaining, but first-half substitute Smith did well for City as he won the ball back with a well-timed sliding challenge on Umeh.
Joe Whitworth had to make his first real save of the match in the 72nd minute, when centre-back Alleyne drove forward and struck well from distance, diving at full-stretch and pushing the ball around his left-hand post.
The home side had an opportunity to get themselves back into the game in unusual circumstances in the 83rd minute as they were awarded an indirect free kick on the penalty spot, after Whitworth was penalised for holding onto the ball for too long.
Palace got the game’s third and final goal as Imray capped off an impressive performance, getting forward from right-back and driving a rasping shot into the top-right hand corner from just inside the penalty area.
The win sees Palace move to 11th in Premier League 2, above the likes of Manchester United and Chelsea.
City: Grant (GK), Samuel, Simpson-Pusey (Smith, 41), Alleyne, Galvez, Susoho (Dunbar-McDonald, 85), Ndala (Oboavwoduo, 52), Adam, Heskey, Taylor, Hamilton (Fapetu, 85).
Sub not used: Murray-Jones (GK)
Palace: Whitworth, Imray, Adaramola, Rodney, Grehan, Sheridan, Wells-Morrison, Ozoh, Ola-Adebomi, Mathurin (Raymond, 77), Umeh (Akinwale, 86).
Subs not used: Izquierdo (GK), Watson, Reid.