Fresh from hitting Ipswich Town for five last weekend, United made a fast start and almost went ahead inside eight minutes when Bruno Fernandes struck the crossbar following a knockdown from Mbeumo inside the area.
It was the 24th time United had hit the woodwork since the start of last season, more than any other Premier League side.
But that did little to dampen United's mood as they continued to put pressure on Everton, repeatedly testing their defence with attacks down both flanks.
Everton were still adjusting to the loss of Iliman N'Diaye, who had joined to Manchester City in a deal worth up to £65m only days earlier, and lacked a consistent creative outlet in attack
That was until two minutes before half-time when Thierno Barry, the club's top scorer this season, headed over after a wonderful cross from makeshift full-back Merlin Rohl, and Jarrad Branthwaite forced a save from Lammens.
But the United breakthrough came less than 60 seconds into the second half, with Mbeumo picking the ball up from Kobbie Mainoo before curling a spectacular effort past Jordan Pickford to double his tally for the campaign.
Brennan Johnson almost drew Everton level, while Rashford continued to threaten.
Everton manager David Moyes summoned summer signings Jack Grealish and Hayden Hackney from the bench midway through the second half, and Hackney was involved in the equaliser when he teed up George to rifle home at the near post in the 83rd minute.
United thought they had won it five minute later when substitute Sesko made it two goals in as many visits to Everton by heading home Luke Shaw's cross.
But it was an Everton substitute who had the final say, with Maitland-Niles' strike sending the home fans into raptures.