After a battling start from both teams, Mark Yeates tried to create a chance for Hull early on, but, after neatly forcing his way into the area, his eight-yard shot was blocked by the Palace defence.
Yeates had another effort on goal soon afterwards and nearly found the net as his 20-yard shot curled just wide of the left-hand post after the ball had been deflected en route to goal.
Michael Bridges was the next player to try his luck for Hull as he collected the ball on the edge of the area from Andy Dawson and attempted to curl a shot into the top right-hand corner, but Gabor Kiraly produced a superb save to deny the former Leeds and Carlisle star.
Taylor's side continued to push forward and found their reward on 57 minutes when the ball fell to Cort on the right-hand side of the box. His first-time shot from 20 yards flew past Boaz Myhill before nestling in the bottom right-hand corner of the net to give Palace the lead.
Hull tried to find the equaliser and came close towards the end of the game when Craig Fagan found himself with space on the right-hand side of the six-yard box. He fired the ball across the face of the goal as Jon Parkin threw himself forward in an attempt to connect only to fail as the ball whistled past his feet.
However, City did finally snatch a point with no time left to spare. Yeates floated the ball in and Turner rose to head the ball into the back of net from six yards out despite Kiraly's protests that he had been pushed.