Leon Clarke grabbed a hat-trick but United conceded five for the first time in Chris Wilder's tenure as Blades boss.
Clarke, who scored four in the previous home game, grabbed three but Fulham escaped with the points.
Lundstram was given the nod to replace Coutts in United's midfield, it was manager Wilder's only change from Friday night's victory at Burton. Coutts missed out with a fractured tibia and Hanson was added to the bench.
After a few minutes of cautiousness by both sides, the Blades grabbed the initiative with the first serious incision through the Fulham defence. The midfield trio of Lundstram, Duffy and Fleck were all involved, the latter shooting at goalkeeper Button, who failed to hold and CLARKE was the quickest to react, slotting home the loose ball.
Fulham almost responded instantly but Blackman made a tremendous one-handed save to deny Ojo and the Blades goalkeeper was called upon with his hands and feet as the visitors looked for a quick route back into the game.
Ojo looked an early threat and United were also probing in a high-energy start. Sharp, Duffy and Stevens looked dangerous in one move before Norwood cleared the danger and O'Connell added to the home side's attacks down the left.
However, in the space of three minutes, the Blades went from leading to trailing, around the half hour mark. A mis-hit back-passed by Carter-Vickers put OJO in the clear and he slid the ball beyond Blackman and then a midfield slip handed SESSEGNON the space to cut in from left back and fire into the bottom corner from 20 yards.
Those goals, bookings for Fredericks and Sharp, and O'Connell trying a trick and being fouled by McDonald, livened up the Bramall Lane crowd ahead of the equaliser on 39 minutes.
The move began with a tackle by Lundstram on Ojo and Stevens and then Fleck had touches before CLARKE was played down the left channel, firing under the body of Button from a narrow angle.
The action, though, continued, and Fulham retook the lead before seven minutes of added time was announced, mainly due to a head injury suffered by Fleck earlier in the half. On 43 minutes Ojo got around the back and his cross perfectly picked SESSEGNON's run to the back post - the teenager volleyed home.
The start of the second period coincided with the visitors slowing down the action at every opportunity. Ayite battled with Basham and Carter-Vickers but poked at Blackman, whilst O'Connell tried to attach himself to attacking moves regularly.
Before the hour mark, Wilder made two changes and altered his formation - Brooks and Carruthers were introduced with the aim of opening up a well-organised visiting outfit.
Fulham introduced Kamara and he was influential straight away. His initial run and shot was blocked by Wright but the substitute found space again soon after, laying off to OJO, whose shot took a decisive deflection off Lundstram to beat Blackman.
Wilder's final change was to introduce Hanson for defender Wright for the final 15 minutes and he also almost made an instant impact. Sharp's shot was deflected over and from Fleck's corner, Hanson's header had beaten goalkeeper Button but McDonald kicked off the line.
And to rub salt in the wounds, Fulham broke away and teenager SESSEGNON found space on the left to fire across Blackman, into the far corner for their fifth.
United continued to pour forward, looking for goals. Penalty appeals were turned down when Brooks clashed with Ream but there was further drama. On 86 minutes, CARRUTHERS surged forward and his shot was deflected past Button at the Kop end.
Six minutes of added time was announced, mainly due to Fulham's timewasting, and there was further action. Two minutes in, it looked as though a Blades move had broken down but Sharp crossed to the back post for CLARKE to head home his third.
This time, though, his goals were in vain.
Blades: Blackman, Stevens, Fleck, O'Connell, Basham, Lundstram, Clarke, Sharp, Wright (Hanson 74), Carter-Vickers (Brooks 58), Duffy (Carruthers 60).
Unused: Moore, Donaldson, Stearman, Lafferty.
Fulham: Button, Fredericks, Sessegnon, Odoi, McDonald (Johansen 83), Cairney, Ayite (Kamara 68), Ream, Norwood, Ojo (Fonte 74), Kalas.
Unused: Bettinelli, Kebano, de la Torre, Edun.