Collins, who joined from Leeds in January last year, headed in unmarked only five minutes into the top-of-the-table encounter.
Ryan Flynn curled over a corner from the left to the far post, which Harry Maguire headed back across goal for Collins to nod in from close range.
Striker Richard Cresswell was denied on 37 minutes by a brilliant save from Town keeper Alex Smithies after a shot from Ched Evans rebounded to him.
Town, beaten for only a third time in 55 league matches, hit back when Jordan Rhodes hit a volley against the post after Jack Hunt's cross was half cleared.
Blades skipper Michael Doyle prevented an equaliser by clearing off the line a goalbound Sean Morrison header from a Danny Ward corner.