Scunthorpe took the lead with their first real opening in the League One clash at Bramall Lane.
Mark Duffy ran at the defence before beating Steve Simonsen with a low shot from just inside the area after 31 minutes.
The visitors had Sam Togwell sent off in the 38th minute when he was shown a red card following a challenge on Michael Doyle.
Williamson scored a 58th-minute equaliser, meeting Ched Evans' cross with a volley following good work from Quinn.
Scunthorpe's Jon Parkin had a shot on target which was saved by Simonsen and the lively Duffy had an effort deflected wide.
But Alan Knill's men had to defend for much of the second period and Quinn struck an 82nd minute winner with a long-range shot which took a deflection.
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