Collins rose superbly to head home James Wallace’s left wing corner from close range to give Nigel Adkins his first win as the Blades’ boss in the second minute of time added on.
United had had the better of the second half with substitute Che Adams forcing two superb saves from Morecambe keeper Barry Roche before Connor Sammon and Marc McNulty both went close.
Morecambe proved tough opponents and went closest to scoring in the first half when Lee Molyneux’s 35-yard free kick beat the United wall but came back off the foot of the post.
The Shrimps had other chances with Mark Howard making a fine save to tip away Jamie Devitt’s long-range effort and Alex Kenyon headed just over from a corner.
The home side’s best chance of the second period came when Devitt was played in on goal but made a mess of his final ball and the chance went begging.