Sam Vokes netted the only goal of the afternoon 14 minutes from time to settle a tight clash played in driving rain.
Hull had plenty of chances during the second half before Vokes struck but found Tom Heaton in the home goal in impressive form.
Defeat means the Tigers are just one point ahead of the Clarets, though Steve Bruce’s men remain on top of the table ahead of Middlesbrough on goal difference.
That will be a small crumb of comfort from an otherwise disappointing afternoon in Lancashire, as the club’s six-game winning run since the turn of the year in league and Cup came to a shuddering halt.
The first half had been a tight affair, both teams doing their best in difficult conditions as the rain fell on Turf Moor.
It wasn’t until the half hour mark that a save worthy of the name was needed and then there were three in quick succession.
The first saw Allan McGregor beat away a Joey Barton free-kick. It was a routine stop but one the Scot executed well.
His next save a minute later was out of the top drawer, the City goalkeeper tipping over another Barton effort after the visitors had been opened up by a well worked corner routine.
Hull’s response was to race down the other end and Tom Heaton had to be alert to keep out Robert Snodgrass’s volley.
Just before the break, Michael Keane headed wide for the hosts but, otherwise, that brief flurry of chances round the half hour mark had been the exception to the rule of the opening 45 minutes.
Heaton was the first goalkeeper to be called into action after the restart, Moses Odubajo unfortunate to see his ferocious shot beaten away on 51 minutes.’
The Clarets ‘keeper again saved his side on the hour by diving bravely at the feet of Abel Hernandez after the Uruguayan had been played through by Robert Snodgrass.
Heaton was almost left red faced two minutes later when he slipped when dealing with a Stephen Ward back pass only to recover in time and clear ahead of Hernandez.
The City top scorer could have broken the deadlock 18 minutes from time when picked out by Sam Clucas’ cross but he headed over.
It proved a costly miss, Burnley opening the scoring four minutes later when Vokes fired in after McGregor had only been able to parry a shot from Joey Barton.
The goalscorer was later booked for diving after an outrageous dive in trying to win a penalty but it couldn’t detract from a fine day for the Clarets.
Hull manager Steve Bruce insisted he would not panic after defeat.
Bruce said: “The conditions were awful today, and when it’s like that one mistake is always likely to cost you.
“Unfortunately here it was us who made that mistake, and we’ve lost the game. I didn’t think we deserved to lose, especially when you look at our second-half performance.
“We carved out some great chances, but maybe we just missed that killer instinct in front of goal that you need.
“We’ve lost, but you just have to accept the result and move on. And we will do. We’ve played well in really difficult conditions. It’s been an opportunity missed for us to create a bit of daylight at the top, but I couldn’t fault the players.
“There was loads of heart and endeavour out there, and that from two top-class teams. It was never going to be a classic in weather like that.
“We’re still at the top so we just have to focus on that I guess. We move on to the next game now and try to get another winning run going.”
Burnley manager Sean Dyche, who signed a contract extension on Friday, stressed he too would remain calm despite his team edging closer to the summit.
Dyche said: “I’m delighted because we’ve beaten a very good Hull side. I always knew it was going to be tight, but looking back I think we controlled the quality of the game overall.
“Hull put us under a bit of pressure early in the second half, but other than the (Abel) Hernandez chance with the diving header I don’t think Tom Heaton had a huge amount to do.
“We didn’t really create that golden chance in the first half, but we are always a threat from set-pieces, and I thought we were again here.
“Hull are a very strong defensive side with a real depth of experience, but we’ve managed to breach them one more time than they breached us.
“I’m really pleased for many different reasons. We’re right up there in the hunt, but nothing will be decided yet.
“It was a fantastic finish from Sam. Despite the ball coming slightly behind him he’s shown his class by calmly tucking the ball home.
“It’s a basic technique, but it’s something that Sam has got plenty of and I’m delighted for him.”