Goals from Wayne Rooney, Daley Blind and Juan Mata put the 11-times Cup winners through, with second-tier Derby only briefly threatening a shock when George Thorne equalised shortly before the break at a noisy Pride Park.
Dutchman Van Gaal has been under mounting pressure after a recent poor run of form and shortage of entertainment, but Manchester United were good value for their fourth-round win.
He had lambasted the media on Thursday, accusing them of “sacking him three times” this season and told the BBC before kickoff it was hard to enjoy his job in the current climate.
Manchester United were booed off after losing 1-0 to Southampton in the Premier League at Old Trafford last weekend, which left them fifth and struggling to qualify for the Champions League, but backed by 5,000 fans they produced a more vibrant performance.
Not that it was all plain sailing.
When Thorne equalised against the run of play after 37 minutes Manchester United’s insecurities briefly returned and, but for a superb last-ditch tackle by Chris Smalling to deny Chris Martin, they could have been trailing at halftime.
Van Gaal told the BBC: “We gave their goal away but at halftime I said that it was a good performance, keep it up and we will win. And we did.”
Rooney had put the visitors ahead after 16 minutes with a goal of real quality.
Receiving the ball on the left edge of the penalty area when he looked marginally offside, the England captain cut back on to his right foot and curled the ball past helpless keeper Scott Carson.
Impressive Manchester United striker Anthony Martial should have doubled their lead but his touch let him down in front of goal and the visitors were stunned when Martin’s clever pass was brought down by Thorne who stabbed a shot past David de Gea.
Derby had the first chance of the second half when right back Cyrus Christie’s low cross just evaded Tom Ince, while at the other end Mata was inches away from restoring Manchester United’s lead.
They deservedly went back in front after 65 minutes when Blind met Mata’s centre with a first-time shot and Mata wrapped it up when he converted Martial’s cross.
The rest of the fourth round matches are scheduled for the weekend with holders Arsenal, who are aiming for a third straight triumph, hosting second-tier Burnley on Saturday.