Manchester United defender
The loss at
United failed to create any clear-cut chances and even the arrival of new signing
"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 it expected a tough test from Everton, which pegged it 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
Ferguson has said he does not expect
But Vidic said it could not attribute the 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 has 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 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."
English Premier League
English Premier League champion Manchester United will start the 2013-2014 season at League Cup winner Swansea.