It was also Leicester’s first defeat of the campaign in all competitions as Manchester City loanee Delap’s goal was enough to give Hull their third league win of the campaign.
The closest Enzo Maresca’s side came to an eqauliser was when second-half substitute Abdul Fatawu hit the post.
But it was a result that knocked the promotion favourites off the top of the Championship table, on a day when they had been hoping to stretch their winning run to seven games.
Leicester started strongly and captain Ricardo Pereira sent a shot from inside the area just wide of the upright.
But it was Hull who took the lead after 15 minutes when Delap cut in from the right and delivered a shot which took a deflection off Leicester defender Jannik Vestergaard to keep it out of goalkeeper Mads Hermansen’s reach.
Leicester were frustrated for long spells as Kelechi Iheanacho saw a handball appeal turned down, and Harry Winks floated a shot just over the bar from 20 yards out.
In first half stoppage time Iheanacho saw a shot cleared off the line by Hull midfielder Regan Slate after a mix-up with goalkeeper Matt Ingram. But the Leicester striker had already been flagged offside.
Leicester manager Maresca made a triple substitution seven minutes into the second half, but Hull should have had a second goal soon after when Delap supplied Aaron Connolly in the 18-yard box, but he put his shot over the bar.
One of Maresca’s substitutes then came close to putting Leicester level after 62 minutes. Winger Fatawu, who joined on loan from Sporting Lisbon on Thursday, curled a shot against the base of the post.
But Hull were still looking for a second goal and Hermansen had to parry a Cyrus Christie shot over the bar, before the Leicester goalkeeper saved at his near post from Scott Twine.
Maresca brought on Jamie Vardy with 22 minutes to go, but Leicester still struggled to find a way through the Hull defence.