Marouane Fellaini's second half header from Darron Gibson's corner was the difference between the two sides, although it's fair to say the Reds had David De Gea to thank for keeping the Toffees at bay up until half-time.
Despite an impressive debut from Shinji Kagawa and the introduction of fellow new signing Robin van Persie, United lacked a cutting edge in the final third and struggled to get in behind a dogged Everton defence.
Vidic says the players are understandably disappointed with the outcome and must be more clinical in front of goal in their next outing at home to Fulham on Saturday.
"We knew it would be a hard game," he 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.
"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."
The United skipper lined up alongside midfielder Michael Carrick and winger Antonio Valencia in the Reds' backline because of injuries to four fellow centre-backs, Jonny Evans, Rio Ferdinand,