Thompson knows his tired team may need a win at Stockport on Sunday, their fourth game in eight days, to see off the threat of Birmingham City to their play-off aspirations.
A point would have done some good but was dashed by set-piece plays that exposed a defence weakened by injury to Roger Nilsen on top of absentee captain David Holdsworth and reliable Nicky Marker.
Colin Little bundled in Crewe's first from a corner with eight minutes remaining and Steve Anthrobus powered home a second from a well-flighted Kevin Street free-kick.
If that made defeat likely then Graham Stuart's tame penalty, after a foul on substitute Petr Kachuro by Lee Unsworth, guaranteed their fate would hinge on one more match.
To describe Ian Hamilton's close-range effort as a consolation would hardly be apt in the circumstances. Thompson even revealed that Stuart could miss the next match with a suspected broken hand.
The Blades are clearly being dulled by their demanding fixture list and Thompson said: "The boys are absolutely exhausted. I have got to keep their heads up and keep smiling. It's difficult when you are going for the big prizes.''
The last thing a tired team needs is to face Crewe, bright and lively, but for all their sharp passing they lacked thrust until the late stages. Their fans were not complaining, cheering a lap of honour and singing their own tune of glory: ``Are you watching, Stoke City?''
Crewe Alexandra (4-4-2): Bankole; Bignot, Unsworth, Walton, S Smith; Garvey, Charnock, Johnson, Street; Anthrobus, Little (Lunt, 86). Substitutes not used: Foran, P Smith.
Sheffield United (4-4-2): Tracey; Borbokis, Nilsen (O'Connor, h-t), Sandford, Quinn; Devlin, Wilder, Hamilton, Stuart (Morris, 84); Saunders, Marcelo (Kachuro, 68). Substitute not used: O'Connor.
Referee: P Rejer (Tipton).Reuse content