United may not have been at their most formidable, but back-to-back wins over Chelsea and Arsenal have shown they are a force to be reckoned with.
And Evra thinks things should get even better because the overwhelming feeling after beating Arsenal 2-1 at Old Trafford on Saturday - despite collecting the three points - was one of disappointment at a failure to win more convincingly.
United certainly had the chances after Robin van Persie struck against his old club after just three minutes but it was not until after Evra's 67th-minute header that the scoreline looked comfortable.
But even after Arsenal had Jack Wilshere sent off, United might have faced a nervy finish had Santi Cazorla's injury-time consolation come a little earlier.
Left-back Evra told MUTV: "It has been a terrific week, we are really happy. It is just the expectations are so high.
"You win at Liverpool, at Chelsea and against Arsenal - I don't think we did that last year - and that is why I think the team know we have the potential and the quality to win the league.
"Just why we are not happy is because we should score more goals. In the end it has been a terrific week for Man United, don't get me wrong.
"I think the confidence is really high for everyone in the team and you could see that in the game in the Capital One Cup we lost.
"I feel sorry for them because it was a great effort and in the last second we conceded a penalty - but Manchester United are back."
United hammered Arsenal 8-2 in the corresponding fixture last season and while a repeat of that scoreline would have been harsh on the visitors, they were fortunate to escape further damage.
"That is why it is true some players were feeling in the dressing room like we had lost the game," said Evra, whose efforts helped United produce one of their best defensive displays of the season.
"To only win 2-1 we are a little bit disappointed, not because we don't respect Arsenal but because the expectations are very high."
United turn their attention to the Champions League this week with a trip to Group H rivals Braga.
Sir Alex Ferguson's men came from behind to beat the Portuguese side last month and another positive result would secure their place in the knockout stages.