The Red Devils were held 1-1 by Swansea City on Sunday, with their 18th Premier League outing of the campaign producing their first stalemate.
A share of the spoils in South Wales was enough to keep United top of the table, but their lead over arch-rivals Manchester City has been cut to four points.
Evra believes added importance has now been placed on a Boxing Day clash with Newcastle United and meetings with West Bromwich Albion and Wigan Athletic at the turn of the year.
"I felt that we deserved to win at Swansea and that is why I am really confident for the rest of the games we have coming up over Christmas," Evra told MUTV.
"We have to move on from the draw and focus on our next game against Newcastle. It is important to win and it is important that we keep playing well.
"I think we need to talk about ourselves and not about another team. We face Newcastle at home, let's make sure we get three points. After that it is West Brom and we must make sure we get three points then too.
"This is an important part of the season and it is an opportunity to get a lot of points. I'm not worried, I am really confident for the game coming up at Old Trafford."