Butler, who had scored just four goals in a seven-and-a-half-year career prior to the game, grabbed two in six second-half minutes at Bramall Lane to send the League One Swans through to the fourth round.
Leon Britton added a third from the penalty spot midway through the second period to complete an afternoon of misery for United and Warnock.
But he has only himself to blame as he made eight changes to the team beaten 3-1 at Middlesbrough on New Year's Day, sending out a signal that Premiership survival is all that counts this season.
Admittedly, Warnock went into the game without suspended captain Chris Morgan and striker Rob Hulse.
However, he also opted to rest key players Phil Jagielka, Claude Davis, Keith Gillespie and Rob Kozluk, while Michael Tonge and Stephen Quinn were benched.
There were still 10 players on show who had played in the Premier League this season, yet their bit-part roles produced a disjoined performance barely worthy of a top-flight team.
Warnock, laid low this week through illness that had kept him away from the training ground for a couple of days, probably wished he had stayed in bed.
He watched a wretched first half from the stands before making his way to the touchline for the second, yet his often-inspirational presence failed to galvanise his makeshift side.
When Kevan Hurst, on his full debut and making a first appearance for United for more than two years, produced a woeful back-pass after 90 seconds it perhaps gave rise as to what might follow.
That forced goalkeeper Paul Gerrard into turning aside a low shot from former Barnsley striker Rory Fallon.
It was hardly an inspiring start from Hurst, who has effectively spent the last 18 months on loan at Chesterfield in three spells.
It certainly set the tone for the opening 40 minutes for United as the much-changed line-up failed to find any cohesion, albeit with Swansea failing to take advantage.
Overall, though, the following 30 minutes were abysmal.
The crowd, and in particular the 5,000 Swans supporters, had to wait until the 32nd minute for the next effort as a long-range side-foot strike from Fallon was comfortably picked off by Gerrard.
Butler soon followed up with a 25-yard free-kick that landed on the roof of the net after it had clipped the head of Christian Nade.
Finally, the Blades came to life in the closing five minutes of the half, with Alan Quinn first glancing just over the crossbar a right-wing free-kick from stand-in captain Nick Montgomery.
Two minutes later Hurst had the opportunity to make a name for himself when a loose ball fell into his path inside the area.
However, his shot on the turn was superbly blocked by goalkeeper Willy Gueret, who was at least alert as he had had nothing to do before then.
After the break United's cup hopes fell apart as Swansea tore into them, although it was the Blades who should have broken the deadlock.
After Gerrard had hauled down a dipping 22-yard strike from Butler after just 45 seconds, Nade then teed up Hurst for an 18-yard shot that clipped the outside of the left-hand post.
What followed was clinical and incisive, with Butler scoring his first in the 53rd minute.
A low right-wing cross from Kristian O'Leary was almost turned home by Adebayo Akinfenwa but for a brave block from centre-back Leigh Bromby, who followed up by denying the striker on the rebound.
However, the ricochet fell to Butler to sidefoot home from eight yards.
The free-transfer signing from Hartlepool in the summer added his and City's second in the 59th minute.
This time Butler fired a 12-yard drive underneath Gerrard after the ball had broken into his path following a David Sommeil tackle on Akinfenwa.
After Alan Quinn came close to pulling one back with a downward header that Gueret saved on the line, United's agony was completed in the 66th minute.
Referee Andy D'Urso had little option but to point to the spot when Derek Geary scythed down Butler in the area.
Butler spurned the opportunity of a hat-trick and instead let Britton beat Gerrard from 12 yards.
The scoreline resulted in a chorus of boo's from those United fans who had remained behind come the final whistle.