The Blades recorded a 100th FA Cup win at Bramall Lane and in the process booked a second round berth.
United avoided a potential banana skin against non-league Worcester City thanks to a penalty from Jose Baxter and late goals from Conor Sammon and Kieron Freeman.
Adkins made four changes for clash with the National League North outfit. Long, Freeman, Campbell-Ryce and Sammon all returned to starting berths at the expense of Howard, Flynn, Done and Sharp.
Campbell-Ryce started on the left, with Baxter on the right.
The game's first opportunity dropped to the non-league side and Long was forced to tip over Gater's effort from the edge of the box. It took the Blades 15 minutes to register a chance - Baxter shooting wide - before the home side began to take control.
The catalyst was the opening goal. Baxter fed Adams, who played a one-two with Sammon before being hauled down by Minihan - penalty was the decision and BAXTER slammed into the bottom corner.
Goalkeeper Vaughan produced a brilliant save to prevent United doubling the advantage on 23 minutes, pushing Campbell-Ryce's shot onto the crossbar, and then Adams attempted an acrobatic effort from Baxter's cross but the ball hit the deck and bounced over.
Adams limped out of the action just after the half hour mark, whilst a spell of rain caused additional problems for both sets of players.
United continued to force Worcester back with Campbell-Ryce, who curled just over after a corner was not cleared, instrumental in a lot of the attacking intent. From a breakaway, Sammon and Baxter linked up but the former headed the latter's cross over.
The final chance of the half, however, saw Worcester go close to a leveller but Burton's half-volley snap-shot from 25 yards was straight at Long.
At the start of the second half United won a number of corners, with Collins heading a delivery from Harris wide as the rain began to fall again.
The chances continued. Freeman stung the palms of Vaughan and Hammond volleyed over. Baxter was at full-stretch as he attempted to turn in Scougall's cross but being just one behind, Worcester were a threat and on 64 minutes Burton and Dunkley almost getting on the end of Nti's cross.
Once again, the chances continued, including three in the space of six minutes, starting with Harris curling a free-kick against the crossbar, which then hit the goalkeeper's back and trickled out for a corner. Sammon beat his man on the left but Baxter's subsequent shot was wide and then, from a Freeman cross, a poke by Coutts was saved by Vaughan.
Worcester's resolve, however, was finally broken on 81 minutes. The dangerous Campbell-Ryce, now operating from the right, delivered a low cross and SAMMON lost his marker to turn the ball home from close range.
United concluded the game with a third, although the visitors will argue there was a suspicion of offside. From Campbell-Ryce's slide pass, FREEMAN cut back for Scougall, whose shot was heading wide until the full-back continued his run to slot home with the final notable action.
Blades: Long, Freeman, Harris, Basham (McGahey 85), Coutts (Reed 88), Baxter, Collins, Sammon, Campbell-Ryce, Adams (Scougall 32), Hammond.
Unused: Howard, Sharp, Done, K Wallace.
Worcester: Vaughan, Minihan (Weir 51), Gudger, Thomas, Sharpe, Gater (Dunkley 60), Nti, Deeney, Burton, Jackman, Vincent (Harrad 74).
Unused: Ross, Smith, Waldron, Brown.