The Reds dominated chunks of the match at Old Trafford and twice hit the woodwork through Wayne Rooney and Danny Welbeck, with Ryan Giggs and second-half substitute Adnan Januzaj also going close to scoring when the action was still goalless.
However, with 86 minutes on the clock, the Toffees struck a decisive blow on the counter attack as Bryan Oviedo latched onto Romelu Lukaku's scuffed shot to finish past David De Gea at the far post, condemning United to a fourth league defeat of the season.
Speaking to MUTV, Moyes could not hide his disappointment: "I felt we really got on top of the game the whole night, but nevertheless we needed a bit of good fortune to go our way and we didn't get that. We had some chances and, if we took them, then the game could have changed dramatically. We didn't take them and in the end we lost the game.
"There were a couple of big moments for us to score, one or two opportunities. Everton were always a threat on the break, they broke very quickly against us but I always felt we could just about nick it. But in the end, we didn't do so."
Asked if he was frustrated by the manner in which Everton scored, via a scruffy Lukaku shot across goal, Moyes was diplomatic in his response and keen to focus on United's own wrongdoings.