Everton, who would face Manchester United again in an FA Cup semi-final at Wembley later this month if Louis van Gaal's side win a replay at West Ham United, almost equalised two minutes later when Phil Jagielka headed against David De Gea's bar.
"It was a stolen victory because we did not play well," United manager Louis van Gaal told Sky Sports.
"Everton are a very good team, better away than at home, and you could see it was a very tight game and we were the lucky ones I think," he told the BBC after United's fourth league win in their last five matches.
"We didn't create so many chances -- two I think and so did they -- but we scored one of ours."
The win lifted United one place to fifth above West Ham and on to 53 points, one behind fourth-placed Manchester City.
Everton, who have lost eight times at home this season but suffered only their second away defeat on Sunday, ran United close although, like the home team, they failed to create many chances in a largely forgettable match.
Their manager Roberto Martinez told the BBC: "There were two games within a game. Up until the goal, we were the team with a better tempo and we always looked a threat.
"The goal was a very soft action, it's not up to our standards. The performances were very, very level and we shouldn't have lost the game.
"It was the sort of game that comes down to small margins -- the ball hitting the crossbar is the difference between us getting something out of the game.
"We are a young group with great potential, but our home form hasn't been anywhere near good enough."
United's record scorer Bobby Charlton, now 78, who made his debut for the club 60 years ago, was given a warm salute when he appeared on the pitch before the match as the South Stand at Old Trafford was renamed the Sir Bobby Charlton Stand in his honour.
(Reporting by Mike Collett; Editing by Toby Davis and Ken Ferris)
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