One-time runaway leaders Cardiff have two games in hand and can double their advantage with victory over Leicester on Tuesday night, but Hull are now their closest rivals for the title after capitalising on Watford and Crystal Palace's weekend woes to go second.
Following a listless first-half largely devoid of goalmouth action, Quinn found space on the edge of penalty area and a calm finish in the 66th minute.
It was enough for City to end a run of seven consecutive defeats against Burnley, who came closest through substitute Sam Vokes when the game was goalless.
Quinn flashed a header wide for the visitors with 33 seconds played, while Burnley enjoyed encouragement of their own from Ross Wallace's eighth-minute corner - David Edgar's hooked shot blocked before Alex Kacaniklic smashed the loose ball narrowly past the angle of post and bar from 18 yards.
Snow was falling by the time Wallace, another Clarets player restored after an injury absence, let fly with a long-range effort that Hull goalkeeper David Stockdale did well to push out of the top corner.
City started to settle into a rhythm shortly after the half-hour and James Chester cut out an ambitious Keith Treacy pass to play in George Boyd, whose goalbound shot forced a last-ditch block from Duff.
Burnley were looking like a team in need of half-time, but when Alex Bruce shanked a volley off-target to conclude a sustained spell of pressure in the 44th minute it summed up Hull's efforts to capitalise.
Gedo tested Lee Grant in the 53rd minute, drawing a sharp near-post stop after Boyd rode a couple of challenges in midfield to play him through.
In response Dyche sent on Vokes in place of Treacy and the Wales international striker almost made an immediate impact when he looped a glancing header from Charlie Austin's right-wing cross past the far post.
Steve Bruce went one better in the 64th minute with a double change, Robert Koren and Jay Simpson replacing Bruce Jnr and Gedo, that paid instant dividends.
Koren drifted into space on the right and his cross found Simpson, whose calmly rolled pass teed-up Quinn to finish clinically.
Burnley failed to mount a sustained response, though 26-goal striker Austin might have levelled late on had he not misjudged a header from Vokes' centre.