After 58 minutes with no nets rippling, the South Yorkshire showdown exploded into bedlam, as two strikes from Ben Brereton Diaz either side of Oli McBurnie's tap-in would ostensibly render Joao Palhinha's effort inconsequential.
However, Bobby Decordova-Reid's drive offered the Cottagers a renewed slice of hope, and with three minutes of injury time on the board, Muniz's scintillating scissor kick rescued a share of the spoils for the visitors, who dealt the latest painful blow to their hosts' survival hopes.
While Fulham boss Marco Silva unsurprisingly saw no reason to alter the pack after his side's brilliant beating of Tottenham Hotspur, Anel Ahmedhodzic, Mason Holgate and Vinicius Souza were brought back into the visitors' XI; George Baldock missed out altogether.
Endeavouring to earn a rare feather in the cap away from home, where they possessed the 17th-worst Premier League record heading into the weekend, the Cottagers made the brighter start, albeit without truly testing Ivo Grbic.
The hitherto quiet Bernd Leno was called into action in the 34th minute, but Ahmedhodzic's effort was too weak to trouble the Fulham number one, who was otherwise a spectator in a half that the Cottagers thoroughly dominated.
Brilliant Blades duo make wasteful Fulham pay
A gilt-edged opening for the red-hot Muniz belatedly arrived in the 37th minute, but Grbic managed to tip the Brazilian's close-range toe poke onto the post, before his audacious rabona rebound sailed over the top.
For all of their possession, the visitors very nearly headed down the tunnel facing a 1-0 deficit as Gustavo Hamer lined up a strike on the edge of the box, but the Blades man could only shoot wide before the whistle signalled the end of a half which sprung into life in the final 10 minutes.
The fast-paced action did not let up at the start of the second period, where Muniz rattled the frame of the goal again - this time from a 47th-minute header - and it only seemed a matter of time before the net would bulge.
Indeed, the deadlock would finally be broken in the 58th minute, but it was the beleaguered Blades who took the lead against the run of play, as McBurnie crossed for Brereton Diaz to fire in from a few yards out after a lackadaisical pass out from the back from Tosin Adarabioyo.
However, a Sheffield United goal was the catalyst that Fulham needed to finally find a clinical edge, as just four minutes after Brereton Diaz's opener, an untracked Palhinha met Andreas Pereira's corner to the near post and directed a header over a stranded Grbic into the far side of the net.
Even Palhinha may have been astounded by the space he was afforded in the Blades box - the hosts' defending was nothing short of horrific - in stark contrast to their ruthless attacking performance.
Palhinha's goal would ostensibly spark the Fulham turnaround, but the Cottagers were only on terms for six minutes as Brereton Diaz and McBurnie swapped roles; the former got to the byline and produced a deft cutback for McBurnie to tap into an empty net.
Magnificent Muniz continues prolific streak
Villarreal loanee Brereton Diaz was proving far too hot for Fulham to handle, and the Chilean remarkably extended Sheffield United's lead in the 70th minute of the match, bundling in at the back post from a delicious Hamer delivery - the Blades' third survived a VAR check, while no further punishment was issues for a goalmouth fracas either.
The Bramall Lane faithful thought that their side had notched a scarcely believable fourth through McBurnie in the 81st minute, but his strike was disallowed for offside against Souza, and Fulham made the most of that let-off.
Tom Cairney battled hard to retain possession the right and advanced centrally before playing in Decordova-Reid, who took one touch before finding the bottom corner with a terrific 25-yard effort with four minutes of normal time remaining.
To the home crowd's bemusement, the Cottagers were gifted 14 minutes of injury time to break Blades hearts, but they would need just three additional moments to wipe out the hosts' lead in incredible fashion.
Once again, Sheffield United's inability to defend crosses proved fatal, as Adama Traore clipped a delicate ball in to Muniz, who left Grbic rooted to the spot with a stunning bicycle kick into the bottom corner.
Chris Wilder's men were on the ropes as the clock ticked down, but both sides would settle for a point apiece, one which keeps Fulham in an unremarkable 12th place, while the Blades remain seven points adrift of safety with just nine games remaining to save their Premier League bacon.
Another taxing away day awaits Fulham on Tuesday, when the Cottagers head north to face Nottingham Forest, while an even more daunting road trip is on Sheffield United's menu, as Wilder's side head to Liverpool on April 4.