The Blades created numerous opportunities and twice saw Jamie Murphy have shots cleared-off the goalline, but Elliott's second-half goal proved decisive.
This match, between two teams harbouring genuine ambitions of promotion, began at a predictably furious pace with Murphy, Bob Harris and Ryan Flynn all going close.
City's defence appeared shaky but going forward they posed a far greater threat with Aaron Wilbraham heading them in front after 20 minutes from Mark Little's cross.
Higdon, who had arrived at Bramall Lane from NEC Nijmegen earlier in the week, continued to cause all sorts of problems and so it was no surprise to see him draw the Blades level 11 minutes later when he met Ben Davies' set-piece.
That was the signal for United to move through the gears and, after having a shot cleared off the line by Aden Flint, Stefan Scougall should have edged them into the lead just before the interval but instead headed over the crossbar.
Home goalkeeper Mark Howard, who had tried desperately to claw Wilbraham's effort away to safety, smothered attempts from both Sam Baldock and Luke Freeman at the start of the second half.
But City were not to be denied and an excellent move, started by Derrick Williams and continued by Wilbraham, ended with Elliott firing home across Howard with 18 minutes remaining.
Murphy thought he had secured United a point during a late scramble in the visitors' penalty area, but Flint acrobatically cleared off the line again.
- Sports & Recreation
- Aaron Wilbraham
- Jamie Murphy
- Nigel Clough