The United boss arrived in the east midlands under heavy pressure and intense scrutiny after months of indifferent results, uninspiring performances and reports - denied by the Dutchman - that he offered to quit after last weekend's home defeat by Southampton.
His captain Wayne Rooney quickly settled the nerves with a superb opening goal after 16 minutes but George Thorne's equaliser before half-time might have had Van Gaal worrying again as Derby threatened to add their name to the list of lower league teams, which include MK Dons, Cambridge, Middlesbrough and Sheffield United, to cause United problems in the cups under his tenure.
However, well-worked goals from Daley Blind and Juan Mata in the second half saw United progress in a much more comfortable manner than was perhaps anticipated before kick-off.
United, as they have in most matches this season, dominated possession early on in the fourth-round encounter
It was not quite the laborious fare often served up by Van Gaal's team in recent months, but United did not muster an effort at goal nonetheless.
However, they made their first serious attack count to devastating effect
Rooney looked marginally offside when he received Anthony Martial's pass on the left side of the penalty area but his curling strike was sublime, with the ball nestling right inside the far post giving Scott Carson no chance.
It was a rare moment of quality in the first half as both teams struggled in the final third, although Martial looked a threat and might have done better with his final pass on a couple of occasions.
Nick Blackman worked David de Gea with a header midway through the first half as Derby finally began to get a foothold in the game
But the equalising goal was as unexpected as it was brilliant.
Chris Martin chased down a loose ball and hooked a lovely reverse pass to midfield runner Thorne
His first touch to kill the ball just inside the box was perfect, it had to be with defenders in tow, allowing him to steer a half volley past De Gea after 37 minutes.
The second half began in much the same vein as the first, with United dominating and Martial looking menacing.
Blackman sent a curling shot narrowly over the crossbar while at the other end Martial and Jesse Lingard combined well but Mata directed his header, from Lingard's deflected cross, wide from just six yards out.
Mata made amends by playing a part in United's second goal
His pass set up Lingard whose low cross from the right was guided into the bottom corner of the net by the instep of Blind from just in front of the penalty spot.
The 65th-minute strike brought a rare display of emotion from Van Gaal who pumped his fists in celebration - perhaps even relief.
United, in front of their largest away following of the season, sealed victory and a place in the fifth round with seven minutes remaining.
Another mazy run from Martial saw him pull the ball back for Mata who brilliantly guided his left-foot shot into the bottom corner of the net - much to the delighted of the 5,460 travelling fans - and Van Gaal.
Source : PA