Falcao had failed to score in his first three appearances following a loan move from AS Monaco but came good with the second-half goal which also gave United successive league wins for the first time under Louis van Gaal.
Everton, with only one league victory all season, missed a penalty and cancelled out Angel Di Maria’s early goal through the in-form Steven Naismith but were denied any reward by Falcao and the brilliance of goalkeeper David De Gea as United edged towards the top four.
Before kick-off, ex-United midfielder and South Korean international Ji-sung Park was unveiled as a new club ambassador by Sir Alex Ferguson as the legendary former manager stepped back onto the Old Trafford pitch for the
first time in a public capacity since his retirement speech in 2013.
It has been far from plain sailing since Ferguson stood down as United manager but van Gaal avoided a fate that had befallen both Ferguson and his ill-fated successor David Moyes – namely losing to Everton in their first Premier League games at home to the Merseyside club.
There had been 21 years between those losses with Everton’s win here last season underlining Moyes’ struggles which ushered in the arrival of van Gaal who may at last be beginning to see a team take shape.
With Wayne Rooney serving the first game of his three-match ban and vice-captain Darren Fletcher only on the bench, van Gaal made Robin van Persie captain with Juan Mata replacing Rooney in the starting line-up.
Everton were still missing both James McCarthy and Seamus Coleman to injury and having got back from a Europa League trip to Russia little more than two full days earlier not surprisingly found themselves stretched early on as United probed for an early goal.
Falcao could have ended his wait for a goal after just four minutes from a Luke Shaw cross which he met with a diving header saved by Tim Howard before van Persie clipped an effort over the bar from Di Maria’a low cross. When Di Maria and Luke Shaw then again combined well down the left, Falcao dragged a shot wide.
But with 26 minutes gone United’s pressure did pay off as Di Maria put them ahead. A cross from Rafael was only partially cleared by Phil Jagielka and when Mata rolled the ball to Di Maria, the Argentina international calmly curled home his third United goal from 15 yards.
Di Maria then tested Howard with a deflected long-range free-kick before Everton asked their first real question of a United defence which has struggled to convince as Romelu Lukaku got in behind teenager Patrick McNair before shooting wide.
And their best spell of the game was rewarded in first-half stoppage with the chance to level when Shaw was judged to have tripped Tony Hibbert, despite appearing to make contact with the ball, to concede a penalty.
Leighton Baines hit his effort well but not wide enough to prevent David de Gea making an excellent save to keep United in front at the break.
Falcao should have doubled the lead six minutes into the second half but the Colombian, 28, appeared to be trying almost too hard to break his duck as he wasted another chance created by Shaw, slicing wide via a touch off defender John Stones.
That looked a costly miss when Everton then made it 1-1 after 55 minutes, Baines linking with Gareth Barry before crossing perfectly for Naismith to head home his fourth goal of the season.
But Falcao’s persistence finally paid off as he claimed the winner with just under half an hour to go.
Everton, effectively down to ten men as they waited to replace Steven Pienaar with Bryan Oviedo, failed to clear properly.
And although Di Maria’s speculative drive wouldn’t have gone in, Falcao stuck out a foot to divert the ball past Howard from 12 yards to make his mark with what proved to be the winner.
De Gea preserved the victory with two superb late saves from Everton substitute Leon Osman and Bryan Oviedo.