Hull created a good chance in the early moments of the game as Bryan Hughes switched the ball from the left to Richard Garcia who beat his man and fired in a 20-yard strike, but the effort was comfortably saved by keeper Steve Simonsen.
Dean Windass had a chance for the home side moments later, but his neat turn and half-volley from ten yards was palmed over by Simonsen.
Richard Cresswell came within inches of putting Stoke in the lead midway through the half when he curled in a 20-yard right-footed shot which whistled past the right-hand post.
The home side played some neat football soon after which resulted in a good cross from Garcia onto the head of Caleb Folan, but the Tiger frontman couldn't get enough power on the header and Simonsen collected the ball with ease.
Stoke grabbed the opener on 44 minutes when Ricardo Fuller raced down the left and fired in a shot which Boaz Myhill parried. However, the Hull defence failed to clear allowing Delap to slide in and fire the ball into the back of the net from close range.
Garcia then raced in on goal early in the second half but the Hull man skied his shot over the bar from ten yards.
Jay-Jay Okocha came off the bench to a standing ovation on 55 minutes as he replaced Dean Marney to make his debut for Hull.
Hull equalised on 87 minutes when Sam Ricketts floated the ball into the box from the right. Folan headed back across goal and substitute Livermore pounced to head past the oncoming Simonsen and level the scores.