The Red Devils enjoyed a majority of possession at Goodison Park but were unable to convert that into goals as Marouane Fellaini's header won the opening-round points
A sustained period of late pressure came to nothing for Sir Alex Ferguson's men as Marouane Fellaini's lone goal made the difference in a first-game loss for last season's Premier League runners-up.
"We knew it would be a hard game,” club captain Vidic told MUTV afterwards. "They did what they're best at - playing long balls and focusing on set-pieces. They had a few chances and in the end deserved to score, but we're never happy when we concede from a set-piece.
"We had some possession but we weren't clinical. When you go away from home you expect the home side to have some chances, but we need to be more clinical."
Fellaini proved a constant aerial threat to the United back line, in which Vidic was partnered by midfielder Michael Carrick due to a spate of injuries, but the Serbian does not feel that short-handedness was to blame for the result.
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"When you come to places like Everton who will play long balls into the box you need big, strong defenders, but we won't use injuries as an excuse," he continued.
"I thought the players who came in gave their best and hopefully we'll have a couple of defenders ready for the next game.
"Towards the end of the game Everton looked a bit tired and I believed, even up until the last minute, we would score because we had some good possession and they sat back a bit. Unfortunately we just couldn't get the goal.
"It wasn't a great game but we're already looking forward to the next one. We've lost our first match [of the season] in the past and finished well so hopefully we can win our next game and go on a long run."
United now prepare to host Fulham on Saturday.