Antonio Valencia gave the hosts an early lead with just four minutes on the clock, with the Toffees then managing to hold their opponents for 80 minutes before a late spate of goals made the result decisive.
Koeman's side have now lost their last three Premier League games but have endured a tough run of fixtures that has seen them play Manchester City, Chelsea, Tottenham Hotspur and United in succession.
"That's football. Sometimes in football you get more than you deserve. Today we deserved more than we got finally," the Dutchman told Sky Sports News . "Ashley Williams made a big mistake for 1-0. After that it was over.
"What I saw with the team today makes me really happy, more happy than last Thursday (a Europa League defeat to Atalanta). After half time, it was a difficult start, fast in the game, but then we grew in the game and our confidence. Not many teams will get those two big chances that we got after half time today. We did everything to score the goal, we didn't, and then it's 2-0.
"It's not about ambition, it's confidence of players and of a team. I try to shout to them to take a higher position. That's something that is in the head of players after a tough period. It's really difficult. As a manager you try to help them, to support them, but when the whistle starts [the game] it's all about you and your teammates. We started too afraid, we did not press.
"But we did not deserve this result. In football you need to be clinical and United showed us today how clinical you need to be."
The result leaves Everton in the relegation zone of the Premier League table with a points tally of just four from their first five fixtures.