Marney clinically despatched home a ball into the area from fellow midfielder Ross Wallace just before half-time at the KC Stadium on Saturday, and the visitors withstood almost constant second-half pressure to bounce back from two straight defeats.
Steve Bruce's Tigers, who had won six of their previous eight league games, dominated after the break, but found goalscoring chances hard to come by against an organised defence.
Hull handed a debut to goalkeeper David Stockdale, on loan from Fulham, while defenders David Edgar and Ben Mee, plus Wallace returned for Burnley.
Stockdale was quickly involved when he saved Burnley striker Martin Paterson's shot from outside the area at full stretch in the sixth minute and at the other end, Hull's in-form Sone Aluko pulled his 20-yard effort off target.
Marney flashed a shot wide from 20 yards, defender Jason Shackell's close-range effort was blocked by opposite number Abdoulaye Faye and Wallace forced Stockdale into another smart save with an angled shot from 30 yards as Burnley enjoyed the better chances.
Mee's volley whistled over the crossbar and Marney's 25-yard shot was well held by Stockdale before Hull responded through midfielder Stephen Quinn, who failed to keep his volley down.
It was the visitors who made their pressure pay to take a 40th-minute lead. Wallace picked out Marney on the right side of the penalty area and the latter buried a right-footed shot into the bottom corner from 12 yards.
Hull winger Robbie Brady came on for Seyi Olofinjana during the interval and defender Alex Bruce was withdrawn injured and replaced by on-loan Sunderland midfielder David Meyler.
Hull wrestled back a more equal share of possession at the start of the second half, but Burnley almost doubled their lead through Shackell, whose header from Wallace's 59th-minute corner was just wide.
German striker Nick Proschwitz replaced Hull defender Liam Rosenior in the 69th minute and volleyed over the crossbar three minutes later before Burnley sent on substitute strikers Sam Vokes and Danny Ings for midfielders Wallace and Brian Stock with quarter of an hour remaining.
Hull continued to pile on the pressure, but chances were few and far between and midfielder Robert Koren's 20-yard shot was comfortably held by Burnley goalkeeper Lee Grant.
Aluko lashed a 30-yard effort wide in the 84th minute as the Tigers became more desperate and the former Birmingham striker pulled another effort from inside the area wide in the 90th minute, but it was not to be for Steve Bruce's side.