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.
United failed to create any clear-cut chances and even the arrival of £24million signing Robin van Persie for the final 22 minutes failed to help them make a breakthrough.
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.
"We had some possession but we weren't clinical," the Serbia international told MUTV.
"When you go away from home you expect the home side to have some chances but we need to be more clinical.
"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."
Vidic was the key player in a patched-up back-four which saw midfielder Michael Carrick playing alongside him in the centre and winger Antonio Valencia at right-back.
Ferguson has said he does not expect Jonny Evans to return for a fortnight while Rio Ferdinand, Chris Smalling and Phil Jones are also sidelined with injury.
But Vidic said they could not attribute their defeat to that.
"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 said.
"I thought the players who came in gave their best and hopefully we'll have a couple of defenders ready for the next game."
The 30-year-old told BBC Sport: "We had some trouble from set-pieces and in the end that was what caused the problem.
"I thought we played some good football but were not clinical and when you are 1-0 down it is difficult to come back against a side like Everton.
"The next game is at home against Fulham, who won their first game 5-0 and looked in good shape.
"But we are playing at home and we have five days to prepare and show character in the game."
United have strengthened their defensive ranks after Vitesse Arnhem left-back Alexander Buttner signed a five-year contract.
The 23-year-old, part of Holland's preliminary squad for this summer's European Championship, joins the side with the manager hoping he will provide the long-term replacement for Patrice Evra, whose future has been the subject of speculation all summer.
"Alexander is one of the best young left-backs in Europe and we're delighted to sign him," Ferguson told manutd.com.
"He's someone we've been watching for a while now. He gives us some really exciting options in that position."
Buttner added: "Joining Manchester United is one of the best days of my life.
"It's a team I have admired for many years now and I can't wait to join my team-mates and play my part in helping the club be successful."
United confirmed Chilean teenager Angelo Henriquez has been granted a work permit which will enable him to complete his move to the Red Devils.
The 18-year-old Universidad de Chile striker underwent a medical with the Red Devils last week and has now been given the green light to join the Barclays Premier League outfit.
A United statement read: "Manchester United is pleased to announce that Angelo Henriquez, 18, has been granted a governing body endorsement which will allow him to register and play for the club.
"He will play his final game for Universidad de Chile tomorrow and a further announcement on him joining United will be made in due course."