The Championship side were made to work hard by Vale after substitute Luke Rodgers had cancelled out Lee Hendrie's opener.
Quinn's stunning goal tipped the scales with Danny Webber adding a third from the penalty spot with minutes to go.
The Championship side went close in the opening minutes when new £2million signing Darius Henderson headed a David Cotterill corner narrowly wide.
The breakthrough almost came in the 14th minute when full-back Derek Geary pumped in a low cross. Ross Davidson's clearance fell nicely for Quinn but his fierce strike deflected round the post.
The Sheffield side swept ahead in the 40th minute when Cotterill found Matt Spring in space. He unselfishly laid the ball back to Hendrie, who curled his right-foot strike from 16 yards beyond Joe Anyon into the far corner.
Vale were level seven minutes after the re-start when Louis Dodds' cross from the right was coolly tucked beyond Ian Bennett's reach by Rodgers at the near post.
Quinn added the second in the 71st minute as Henderson's header was cleared straight to the little midfielder, who drilled the ball into the roof of the net from 15 yards.
Marc Richards almost took the game into extra time in the dying minutes with a stinging drive from 22 yards which was tipped over the top by Bennett .
But Webber added a third from the penalty spot a minute from time after defender Sam Stockley had brought down Cotterill.