The Blades progressed to the Johnstone's Paint Trophy second round thanks to a penalty shoot-out victory on Tuesday evening.
Captain Jay McEveley scored the winning penalty after two shoot-out saves from George Long at Hartlepool United.
Inside 90 minutes, Ryan Flynn gave United an early lead but Scott Fenwick levelled for the hosts just after the break.
Kennedy and Flynn made their first appearances of the season as boss Adkins named a total of nine changes with only McEveley and Basham remaining from Saturday's starting line-up.
Wearing the green away strip, United made the brighter start. Basham almost put Flynn through and Woolford, in the clear, had a shot turned around a post by goalkeeper Bartlett before the visitors opened the scoring on seven minutes.
McNulty laid off to Woolford, whose effort was saved by Barlett but the loose ball broke to FLYNN, on the left, and he brought the ball back inside with his first touch before slotting home.
Woolford had a third effort saved by Bartlett on 13 minutes as United's dominance continued, but the chances dried up, whilst at the other end Long was barely troubled.
It took until 40 minutes for the home side to register a serious chance but Carroll headed wide Bingham's cross. Hartlepool did come close to levelling just before the break when right-back Duckworth cut in and curled a left footer just wide of the far post but it worried Long into a full length dive.
Hartlepool's momentum continued early in the second half and they were rewarded with a goal to level the scores. United squandered possession in midfield and following Bingham's burst, FENWICK was in the clear to beat Long.
Basham's driving run saw Woolford sidefoot just wide as United looked to wrestle back the impetus and just after the hour McNulty was crowded out by Duckworth as the Blades striker looked to get a shot away.
Carroll let fly with a 25-yarder that was just wide on 70 minutes and for the Blades, McGahey's cross caught out Nearney but Woolford's shot was saved by Bartlett's legs as chances were exchanged once more.
With five minutes to go, Adkins showed his intent to win the tie by replacing defender Kennedy with striker Sharp but a winner was not forthcoming and, as in all JPT ties, the game went straight to penalties to decide the winner.
PENALTIES (Hartlepool first)
Bingham - scores - 1-0
Higdon - scores - 1-1
Oates - scores - 2-1
Sharp - scores - 2-2
Harrison - saved - 2-2
McNulty - scores - 2-3
Fenwick - saved - 2-3
Woolford - saved - 2-3
Featherstone - scored - 3-3
McEveley - scored - 3-4
Hartlepool: Bartlett, Carroll, Worley, Harrison, Bingham, Oates, Featherstone, Fenwick, Hawkins (Nelson-Addy 90), Jones, Duckworth (Nearney 65).
Unused: Oyenuga, Smith, Denton.
Blades: Long, Basham, Flynn (Campbell-Ryce 72), McNulty, Woolford, McEveley, Alcock, Higdon, Kennedy (Sharp 85), K Wallace, McGahey.
Unused: Howard, Collins, Sammon.