The one-time runaway leaders 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 in the 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.
One win in nine games makes grim reading for Clarets boss Sean Dyche, whose shot-shy side have seen their play-off hopes peter out.
Dyche made five changes to the Burnley side that drew 1-1 with Barnsley six days ago, including recalls for fit-again defenders Michael Duff and Ben Mee.
Paul McShane came into the Hull XI, with joint-top scorer Robert Koren dropped to the bench, while Alex Bruce moved from the back three to a defensive midfield role on his 250th career appearance.
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.
From the resulting set piece, Chris McCann rose higher than the visitors' defence but could only head over.
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.
Quinn's speculative strike from distance early in the second period presented home goalkeeper Lee Grant with a routine save and his first action of the evening.
Gedo handed Grant a more thorough examination 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, 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, Vokes seeing a first-time effort deflected behind by Chester in the 75th minute, and when Austin hesitated in a promising attacking position by turning back and hitting a pass out of play the locals showed their displeasure.
The 26-goal striker has scored his side's only two goals in the past six games and might have levelled late on had he not misjudged a header from Vokes' centre.