The Colombia international striker, on a season-long loan from Monaco with a view to a permanent deal, crucially converted the outstanding Angel di Maria's wayward effort in the second half as United claimed successive victories for the first time under Louis van Gaal.
Di Maria had put the hosts ahead with his third goal in five games before David de Gea dived low to his right to keep out Leighton Baines' penalty on the stroke of half-time.
But, after Steven Naismith levelled, Falcao popped up with a timely first goal for the club in the 62nd minute, while team-mate De Gea produced a string of superb saves to deny Everton a point.
The result was a huge boost to United ahead of a two-week international break, during which Van Gaal will hope some of his injured players – including Luke Shaw following his 70th-minute withdrawal – will regain full fitness.
Defeat for Everton, meanwhile, extends the Merseyside club's winless streak to five games in all competitions.
Juan Mata deputised for the suspended Wayne Rooney, with Robin van Persie taking the captain's armband, and Antonio Valencia came in for injured midfielder Ander Herrera.
The visitors also made two alterations from last weekend's 1-1 draw at Liverpool, Steven Pienaar and Aiden McGeady replacing Kevin Mirallas and James McCarthy.
United began confidently in attack, Falcao forcing an early save from Everton goalkeeper Tim Howard with a diving header from Shaw's cross.
Van Persie was next to threaten when turning an effort just over following a powerful Di Maria centre.
The hosts continued to look the more dangerous, even if Everton were enjoying the lion's share of possession.
Roberto Martinez's side found themselves behind in the 27th minute, Di Maria continuing his fine form since moving to Old Trafford from Real Madrid in August.
After Phil Jagielka had brilliantly hooked Rafael da Silva's cross clear, the ball fell to Mata, who squared for the Argentina international to curl beyond Howard.
Howard was called into action again soon after, turning Di Maria's deflected free-kick over the crossbar.
United were fortunate to head into the break ahead, though, with De Gea denying Baines from the penalty spot in first-half stoppage time after Shaw had felled Tony Hibbert.
Van Gaal's men were pegged back in the 55th minute when Naismith met Baines' superb cross with a clever diving header to haul the sides level.
Suddenly in the ascendancy, Everton pressed forward further with Jagielka seeing a header cleared off the line after meeting Baines' corner.
But United found a response, Falcao producing a smart finish to convert Di Maria's effort.
Nervy moments followed for the home side, De Gea denying Leon Osman with a stunning one-handed stop.
De Gea's heroics did not end there, the Spaniard once more producing a stunning save to turn over Bryan Oviedo's volley after Tyler Blackett had superbly blocked Gareth Barry's strike.