The loss at Goodison Park was the first time since 2008 the Red Devils had failed to win their opening match of the season and the first time in eight years they had lost at the start of a campaign.
Manager Sir Alex Ferguson was unhappy with their lack of a cutting edge and their failure to get Van Persie in the game and Vidic agreed, telling MUTV: "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."
He went on: "It wasn't a great game but we're already looking forward to the next one.
"We've lost our first match in the past and finished well so hopefully we can win our next game and go on a long run."
Vidic said they expected a tough test from Everton, who pegged them back to a 4-4 draw in April to damage United's title aspirations, but failed to combat the Toffees' strengths.
"We knew it would be a hard game," he added.
"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.
"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."