The Bramall Lane side survived a couple of early scares before Quinn struck in the 29th minute and Gray killed off Burnley on 49 minutes, as he netted his 14th of the season.
Burnley should have snatched the lead in the ninth minute when midfielder John Oster picked out Graham Branch at the far post, but from six yards he powered his header wide of the target.
Ade Akinbiyi, set to move to Bramall Lane for £650,000 earlier this month before deciding to join Burnley, almost snatched the opener minutes later.
The big striker, back after a three-match ban after he was red carded on his debut against Sunderland, turned quickly in the box and his shot beat Paddy Kenny but spun wide of the far post.
The Sheffield defence dithered again as Branch steamed towards the box and as a gap opened he shot from long range, only to slice his low drive high and wide.
Michael Tonge carved out United's first opening for Jon Harley but the ex-Fulham full-back pulled his shot over the bar.
Burnley's probing almost paid off after 26 minutes, Branch feeding Mo Camara whose low strike was deflected into the side netting by Andy Liddell.
United broke the deadlock three minutes later as Clarets keeper Brian Jensen blundered.
Full back Rob Kozluk was the architect with a curling cross from the right flank. The Burnley keeper charged off his line but palmed the ball straight to the waiting Quinn who squeezed the ball into an empty net from seven yards.
Oster shot wide from a Camara cross before Jensen produced two blocks in quick succession to keep out Steven Kabba as the Blades pressed for a second.
The goal came four minutes after the re-start, Quinn threading the ball to Gray who powered a low shot beyond Jensen from 16 yards.
Akinbiyi pulled one back in the 84th minute, firing in substitute Ian Moore's cross, but it counted for little.