It put United fourth in the table on 11 points, eight behind leaders Chelsea who beat Arsenal 2-0 on Sunday.
               
United manager Louis van Gaal was able to celebrate a second successive victory for the first time since taking charge in the close-season, but as in the recent 2-1 win over West Ham United, his team almost threw two points away at the end.
               
Van Gaal admitted being grateful to goalkeeper David De Gea for a series of fine saves, including one from Leighton Baines's penalty and then three more in the last few minutes.
               
"When you hold a penalty and then hold three magnificent saves in the dying seconds, then you are great because you win the game," the manager said.
               
He also praised Angel Di Maria, who scored the first goal and contributed the assist for Falcao's winner but added: "It's strange because every time the first half is very good and then the second half is so different."
               
Argentina international Di Maria scored his third goal in four games to give United the lead in the 27th minute but Baines, after missing a penalty for the first time with his 15th Premier League spot-kick, crossed for Steven Naismith to head an equaliser in the 55th minute.
               
Colombian striker Falcao, playing in place of United's suspended captain Wayne Rooney, poached the winner in the 62nd minute from a mis-hit shot by Di Maria.
               
In the last few minutes De Gea made three outstanding saves, twice from Leon Osman and once from Costa Rican Bryan Oviedo as United's inexperienced defence struggled to hold on.
               
Di Maria suggested that the goalkeeper should have been named man of the match by broadcaster Sky Sports instead of him.
               
He said he was delighted too for fellow South American Falcao in scoring his first goal for the club, adding: "It's great for him, he's been looking for it for a while and it's very important for him.”
               
"Not only him but the whole team is on the right road now,” he added.
               
Everton were left in the bottom four with their manager Roberto Martinez ruing De Gea's brilliant performance and complaining that the referee should have stopped play before United's second goal because Steven Pienaar was down injured.
               
"The referee made a major error there," he said.

"Our second-half performance deserved more than we got and we finished really strong at the end,” he added.

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