He burst clear of the Blades defence and gave keeper Jamie Anderson no chance with a clinical finish.
Nicky Law went close to opening the scoring for the visitors after 12 minutes when he saw an effort come crashing back off the crossbar, while Andy Keogh also went close for the Iron.
Rookie Iron keeper Josh Lillis pulled off a brilliant stop to deny Ade Akinbiyi just before the break.
Ian Baraclough saw a deflected effort tipped away by Blades keeper Ian Bennett, who was later replaced by Anderson.
The game lost its urgency in the second-half after a raft of substitutions, before Asamoah's late strike.
Asamoah also saw a late stoppage-time effort tipped wide of the post by Anderson.
Trialist Kyle Nix came off the bench for the last 16 minutes, but made little impact after replacing Dave Mulligan.