The Toffees were enjoying a productive passive of play before Marcus Rashford and Timothy Fosu-Mensah combined to release Anthony Martial for a 54th minute decider at Old Trafford.
Phil Jagielka hit the crossbar shortly afterwards but the visitors were unable to prevent their first Premier League defeat on the road since October, leaving them 12th in the standings with 38 points from 30 games.
United moved within one point of fourth-placed neighbours Manchester City with the result, intensifying the race for a Champions League berth.
"There were two games within a game," Martinez told BBC Sport. "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's that sort of game that came down to small margins - the ball hitting the crossbar is the difference between us getting something out of the game."