The visitors created the better first-half chances with Marney flashing a shot wide early on.
Clarets defender Jason Shackell's close-range effort was blocked by opposite number Abdoulaye Faye and Ross Wallace forced David Stockdale into another smart save with an angled shot from 30 yards.
The visitors made their pressure pay to take a 40th-minute lead when 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.
The hosts dominated after the break, but found goalscoring chances hard to come by against an organised defence.
Sone Aluko lashed a 30-yard effort wide in the 84th minute as the Tigers became more desperate, but it was not to be for Steve Bruce's side.