Hull took the lead early on as Nick Barmby played a neat ball through to Fagan, who raced into the box from the left and cleverly set up Marney, who sent the ball into the back of the net with a sweet right-footed effort from ten yards.
Hull were then awarded a penalty on 23 minutes as Barmby was brought down by Jon Harley in the box. Fagan stepped up to take the penalty and calmly slotted the ball into the bottom corner to double the home side's lead.
Clarets boss Steve Cotterill made a double substitution on 30 minutes as Steve Foster came off for John McGreal and James O'Connor was replaced by Kyle Lafferty, in a clear attempt to get Burnley back into the game.
Gifton Noel-Williams came close to grabbing a goal for the visitors before the break with a glancing header from a Wade Elliott corner which bounced just wide of the post.
Noel-Williams had another chance early in the second half, but his sharp left-footed effort from ten yards was well saved by Tigers goalkeeper Boaz Myhill.
Hull had a great opportunity to go further in front midway through the half as Andy Dawson found Jon Parkin unmarked in the box with a great cross from the left, but the big frontman failed to finish it, as he slammed the ball straight at Brian Jensen, who managed to palm the effort away.
The home side came even closer later in the game as Marney whipped in a low cross from the right with Fagan getting on the end of it.
However, his shot from close range slammed against the crossbar and Burnley were able to clear the danger.
The victory lifts the Tigers to within two points of fourth bottom Barnsley, with caretaker boss Phil Brown enhancing his claims for the City job on a full-time basis.
Burnley are now marooned in mid-table, with their wafer-thin squad beginning to show the strain in the bust festive programme of games.