United started brightly as Falcao and Robin van Persie threatened in the early stages at Old Trafford, before Angel di Maria fired them into a 27th-minute lead, curling into the bottom corner after being found inside the area by Juan Mata.
Van Gaal’s side were in control for the majority of the first half, but they had David de Gea to thank for preserving their lead at the break as he denied Leighton Baines from the penalty spot after Luke Shaw had hacked down Tony Hibbert.
Everton did manage to get on level terms 10 minutes after the restart as Baines found Steven Naismith unmarked inside the six-yard box which a superb delivery to head beyond the helpless De Gea.
But United found themselves back in front just seven minutes later as Di Maria’s strike from range fell kindly into the path of Falcao, who was able to guide it past Tim Howard for his first goal since arriving from Monaco this summer.
De Gea then produced stunning late saves to deny substitutes Leon Osman and Bryan Oviedo as United were made to hold on for the three points.
Van Gaal will face tougher tests after the international break – Everton were ravaged by injuries and clearly tired after a 4,000-mile trip to Krasnodar – but the United boss will be happy to have chalked up his third win, especially as it came without suspended captain Wayne Rooney.
The only negative for the Dutchman was that Shaw hobbled off in the second half with an injury.
United’s play at the start had all the hallmarks of a typical Red Devils performance under former boss Sir Alex Ferguson. The hosts flew at Everton from the first whistle with some high-tempo play that was easy on the eye.
Shaw found Falcao with a perfect delivery in the fourth minute, but the Colombian mis-timed his header, which flew wide.
Di Maria then created a yard of space on the left wing and sent over another perfect cross, but Robin van Persie blazed over the bar.
Falcao wasted a chance in front of goal soon after following a lovely move which included a majestic mid-air flick from Di Maria.
Marcos Rojo frayed a few nerves with a couple of risky passes, but otherwise United looked strong at the back thanks to a solid display from 19-year-old Paddy McNair.
After applying heavy pressure, United finally broke the deadlock in the 27th minute. Phil Jagielka attempted to hook Rafael’s cross clear, but instead it landed at the feet of Mata, who squared to Di Maria and he tucked the ball beyond Tim Howard with a curling right-footed shot.
Di Maria would have added a second straight away had Howard not saved his deflected free-kick with a superb diving save.
Romelu Lukaku, isolated up front for Everton, took down Baines’ long pass on his chest but volleyed wide before Shaw put United’s good first-half work in jeopardy by clumsily bringing Hibbert down in the box from behind.
Friend pointed to the spot, but Baines saw his impeccable record vanish before his eyes as De Gea flung his right arm out to push the penalty wide.
United continued to press after the restart, but they failed to add to their goal tally immediately.
Falcao was the most guilty party. The on-loan Moncao hitman somehow failed to turn in Di Maria’s flat ball into the box and he then shinned Shaw’s cross wide from eight yards out.
Everton then stunned the home crowd with an equaliser as Baines made up for his penalty miss by swinging a deep cross towards Naismith, who powered his header past De Gea after peeling off his marker Shaw.
Everton sensed blood and looked for a second. They almost got it when Jagielka headed Aiden McGeady’s cross goalwards, but Falcao cleared off the line.
Falcao then finally found the net at the other end to put United ahead. The Colombian timed his run inside the box to perfection and prodded in Di Maria’s mis-cued shot from close range.
United then suffered a blow when Shaw hobbled off with an injury. Tyler Blackett came on in his place and Van Gaal also introduced James Wilson for Falcao, who received a standing ovation for his efforts.
Osman broke through the United defence in the final minute, but once again De Gea rescued the hosts with a fine diving save.
The Spaniard denied Oviedo with another acrobatic save and the match ended on a bad note for Everton when John Stones was carried off on a stretcher with an injury.