Sheffield United and Aston Villa each climbed a place in the Premiership table after an entertaining draw.
Quinn's equaliser set up an end-to-end second half
Villa came out of the blocks fast with Stiliyan Petrov slamming in a first goal for the club within two minutes.
Chris Sutton also hit the bar but Villa failed to take advantage of lacklustre hosts who bounced back after the break.
Stephen Quinn's volley from a long throw drew the Blades level and Danny Webber hooked in a second, but within seconds Milan Baros pounced to level.
It was a fair result in a match that epitomised the cliche of a game of two halves after a remarkable turnaround from the hosts.
They fell behind on 76 seconds when the recalled Sutton linked up on the edge of the area with Petrov, and after the striker's initial shot was blocked by Leigh Bromby, the Bulgarian struck.
More than a quarter of the game had gone before United found their feet and belatedly pushed forward with anything approaching menace.
But that only buoyed Villa and the verve of their play suggested a first win in a month was there for the taking.
Sutton, back from a calf injury, headed a floated Petrov free-kick on to the crossbar and but for a fine double save by Paddy Kenny from Gary Cahill, Villa would have doubled their lead following a teasing Petrov cross.
Having failed to muster a shot on goal of any significance in the first half, United took a leaf out of Villa's book and took the game to their visitors from the off after the interval.
Quinn's first senior goal for United, after Liam Ridgewell mis-directed a header from a long Rob Kozluk throw into his path in the box, signalled a change in momentum.
Suddenly the Blades had found a cutting edge and looked the more likely victors.
Rob Hulse headed powerfully straight at Villa goalkeeper Stuart Taylor from Keith Gillespie's corner before Webber put the hosts ahead.
United worked a short corner before sending in a deep cross, Chris Morgan climbed above Gavin McCann and Webber hooked the captain's header in.
But Bramall Lane was still celebrating when Baros thundered a shot beyond Kenny from close range 44 seconds later.
Steven Davis clipped a ball into the box, Sutton drew two defenders and Baros found himself in an ocean of space to pick his spot and crack in the flick-on from his strike partner for a first goal in eight months.
That lack of concentration at the back ultimately cost United a third successive Premiership victory.
The tempo of the game continued to increase and both teams had chances to find a winner, but there was to be no fifth goal in a topsy-turvy, end-to-end match.
Hulse and substitute Christian Nade saw efforts blocked in an almighty goalmouth scramble after Taylor had initially saved well low to his left.
And Quinn forced a fine block from Taylor, who ended the game limping, with a rasping shot from the angle.
At the other end Gabriel Agbonlahor wasted a chance before Petrov had Kenny scrambling across his goalmouth with an injury-time free-kick which whistled wide.
However, an enthralling encounter ended on a sour note when Villa substitute Juan Pablo Angel blatantly elbowed Morgan in the box.
Villa's seventh draw in nine away games meant they missed out on the chance to climb into the top five, while the Blades are now three points above the relegation zone.
Sheffield United boss Neil Warnock:
"What a great game it was, but for a manager it's a bit of a nightmare. We just give teams massive lifts.
"We claw our way back into it and then we celebrate and give the ball away.
"A sloppy pass from Webbs on halfway and they end up scoring a goal. I wish I'd taken him off when he'd scored."
Aston Villa manager Martin O'Neill:
"We got ourselves a goal in front and could have been two, three, four up and out of sight. But we weren't able to nail it.
"We were absolutely in control, but then had to rely on another brilliant goal to get something from the game.
"Creatively we were excellent and there were many great things to take out of the game, but obviously we'll have to do better defending the set-pieces."
Sheff Utd: Kenny, Kozluk, Bromby, Morgan, Geary, Gillespie, Jagielka, Montgomery, Stephen Quinn, Hulse, Webber (Nade 68).
Subs Not Used: Davis, Alan Quinn, Armstrong, Kazim-Richards.
Booked: Kozluk, Bromby.
Goals: Stephen Quinn 50, Webber 64.
Aston Villa: Taylor, Mellberg, Cahill, Ridgewell, Bouma (Hughes 67), Agbonlahor, McCann, Petrov, Davis (Osbourne 75), Sutton (Angel 83), Baros.
Subs Not Used: Olejnik, Gardner.
Goals: Petrov 2, Baros 64.
Ref: M Halsey (Lancashire).