The visitors nearly snatched the lead with less than two minutes gone when Hull’s Michael Turner and Palace’s Mark Hudson challenged for Ben Watson’s corner and the ball cannoned against the post.
Dean Windass’ superb 25-yard free-kick then hit the bar.
Hull went ahead after Fraizer Campbell spun 20 yards from goal and drilled a low shot into the bottom-left corner on 18 minutes.
The visitors levelled in the 38th minute.
Wayne Brown won the ball in his own six-yard box but gave possession straight to Scott Sinclair, who stroked the ball into the bottom-left corner.
Ashbee then scored with a tremendous header from 10 yards following a Dean Marney corner on 85 minutes and Hull hung on.