Van Gaal had admitted prior to the fourth-round tie that defeat at the iPro Stadium could cost him his job, having seen his side booed off following last weekend’s abject 1-0 loss to Southampton at Old Trafford.
But after Wayne Rooney’s superb opener had been cancelled out by George Thorne, Blind and Mata struck in clinical fashion to take United into the last 16 and ease some of the pressure on the embattled Van Gaal.
“We gave their goal away, but at half-time I said that it was a good performance, keep it up and we will win. And we did,” said Van Gaal.
“The Premier League is very important, but the FA Cup is the greatest cup in England with a long and important history. We haven’t won it for a long time so we dream of it.”
United’s dream of a first FA Cup win since 2004 remains alive and they will return to Premier League matters, at home to Stoke City on Tuesday, with confidence at least partly restored.
Derby, meanwhile, must wait for their next opportunity to end a wait for an FA Cup victory over United that stretches back to February 1897.
“I don’t think we were unlucky. Manchester United deserved to win the game,” conceded Derby manager Paul Clement.
“For a team that’s supposed to be in disarray and lacking confidence, they played well.”
A defensive injury glut meant that there was an unfamiliar look to the visitors’ starting XI, with Guillermo Varela and 18-year-old Cameron Borthwick-Jackson at full-back, but the sharpness of United’s passing in the early stages belied their recent struggles.
– Striker-like timing –
It took them less than three minutes to cut Derby open, the impressive Anthony Martial side-footing over from Rooney’s lay-off, and in the 16th minute a similar combination yielded the opening goal.
On United’s left, Martial’s pass to the edge of the box was gathered by Rooney, who moved the ball onto his right foot and shaped a sublime shot around his one-time England team-mate Scott Carson and into the top-right corner.
The United captain had been marginally offside when the ball was played to him, but the goal stood, giving Rooney his sixth goal in six games.
Derby, currently fifth in the Championship, had barely laid a glove on United, David de Gea comfortably fielding a 20-yard shot from right-back Cyrus Christie and a downward header by Nick Blackman, but against the run of play they equalised in the 37th minute.
Chris Martin hoisted a clever pass into the box from the right and Thorne, untracked by United’s midfielders, strode through to stab a left-foot shot inside the left-hand post.
The boisterous travelling fans fell silent and Thomas Ince threatened to add to their disquiet shortly before half-time when he cut in from the right and flashed a shot wide.
Mocking chants of “Sacked in the morning!” echoed around the ground and there was another scare for Van Gaal early in the second period when Blackman drilled over.
But United gathered their senses, began to probe — Mata heading wide from point-blank range, Marouane Fellaini heading straight at Carson — and in the 65th minute they drew level.
Having set an attack in motion, Blind continued his run from centre-back and arrived with striker-like timing to turn Jesse Lingard’s low right-wing cross past Carson with an assured finish from 10 yards.
“Daley Blind’s goal? You have to sniff it and he sniffed it,” said Van Gaal. “It was a great goal.”
Michael Carrick made his return from a four-game lay-off as a 73rd-minute replacement for Morgan Schneiderlin and United made the game safe 10 minutes later when Mata tucked away Martial’s cut-back.