Goals from Nani, Darron Gibson, Cristiano Ronaldo and Danny Welbeck ensured that a Miles Addison header was nothing more than a consolation for the hosts.
The Rams were able to contain the Premier League leaders for almost half-an-hour before the deadlock was broken.
Both sides had spurned good opportunities by that stage, but there was to be no stopping a 20-yard thunderbolt from Nani.
In an almost exact replica of the goal he scored against Derby in the second leg of their Carling Cup semi-final meeting last month, the Portuguese winger drifted in from the flank before firing an unstoppable effort into the top corner.
That lead was then doubled a minute before the interval when a Ronaldo free-kick broke kindly for Gibson to steer a controlled volley past the stranded Stephen Bywater.
The result was put beyond doubt three minutes into the second period when Ronaldo rose highest to meet a Ryan Giggs corner and power a header into the back of the net.
Derby did snatch a lifeline when Addison glanced home a Kris Commons cross on 56 minutes, but they were ultimately unable to give the Red Devils a scare by forcing a further breakthrough.
It was therefore left to substitute Welbeck to round off the scoring nine minutes from time as he curled a sumptuous fourth past Bywater's outstretched hand.
ReactionBywater was twice called upon to keep the visitors out during the opening exchanges, first denying Nani with a low save, then pulling off an amazing reaction effort to repel Park Ji-sung.
But it was already pretty obvious Derby needed an unexpected turn of events to emerge victorious and once referee Alan Wiley had rejected Rob Hulse's claims for a penalty when he went down under Rio Ferdinand's challenge, it was always unlikely to come.
Nani seems to have lost his way a little in his second season at Old Trafford, yet he remains capable of those moments of genius for which compatriot Ronaldo is so famed.
And, when he collected Giggs' square pass and raced across the penalty area, the fierce 20-yard shot that followed was nothing out of character as it left Bywater with no chance.
One Kris Commons free-kick that tested Ben Foster was about all Derby could manage in response before a bizarre period followed in which United were denied a goal that should have been allowed, and given one that should have been ruled out.
Ronaldo's celebrations were cut short by a very late flag after he raced clear and slotted the ball past Bywater.
The officials needed to confer before acknowledging Giggs had flicked Ben Foster's long throw through to the Portuguese superstar.
Sir Alex Ferguson was not happy, making his feelings known to fourth official Rob Styles in no uncertain manner.
Fortunately, United's grumbles did not last too long, although how Rafael was not ruled to be offside as he ran straight in front of Bywater as Gibson volleyed in was a mystery.
ElaborateAny lingering doubt over who would be in the last eight was removed within three minutes of the restart.
Giggs was again the provider, this time in routine fashion with a near-post corner which Ronaldo powered home, his elaborate celebration suggesting his 15th goal of the campaign should have come earlier.
To all intents and purposes, it was game over, even if Miles Addison guided home a header to pull one back for Derby almost immediately.
Ronaldo spurned a couple of chances to add to his tally, with Darren Fletcher failing to profit from a rebound.
Enthusiastically Derby pressed forward in reply, Gary Teale bringing an excellent save out of Foster, who was handed a rare start ahead of record-breaking Edwin van der Sar.
But United were too good, with Giggs in particular truly majestic.
At 35, with a barrow-load of trophies already, it could be argued the former Wales skipper is playing more consistently well than at any stage of his career.
It still rankles with Ferguson that in the 1999 treble-winning season, Giggs, along with the remainder of United's squad, was overlooked for all the major individual prizes in favour of David Ginola.
He further enhanced his credentials for this season's player prizes by playing a role in Welbeck's late effort.
1st Half Goals
Shots on Target
Shots off Target