The Red Devils have struggled for consistency following the arrival of David Moyes as Sir Alex Ferguson's successor.
They are still only sixth in the Premier League, ahead of Moyes' former club Everton only on goal difference, and last Sunday dropped two more points by conceding an injury-time equaliser at Cardiff.
Yet Wednesday night's five-goal Champions League mauling of Bayer Leverkusen underlined what United could achieve.
Moyes immediately stated the best is still to come from his side and Evra, a veteran of five Premier League title-winning campaigns, is equally optimistic.
"We are really confident we can do something big this year," said the 32-year-old defender. "There is always that belief.
"Every year is a new challenge. I trust the manager and I believe in this team. If I didn't feel that I wouldn't play anymore."
Although Evra's future has been called into question following Moyes' pursuit of Leighton Baines, and reports have emerged from France indicating big money is available at his cash-rich former club Monaco should he elect to leave Old Trafford this summer rather than sign a contract extension, he always speaks of United with an acute sense of pride.
It is the same now, as he looks ahead to what is, in his words, a must-not-lose trip to Tottenham on Sunday.
"People always write Manchester United off," he said.
"They said the same things last year. In the end we won the league by 13 points.
"Wait until the end of the season. We are strong. We have mentally strong characters.
"Every year is a challenge playing for Manchester United. It is never easy because when you put that shirt on you have the story of the club to consider.
"We have to show we love this club."
Although United have extended their unbeaten run to 11 games, they have still won only half of 12 Premier League fixtures.
Under normal circumstances they would have expected to be further behind than their current position, seven adrift of leaders Arsenal but only three off second-placed Liverpool.
Not that Evra cares too much about that.
"If you calculate it, we drew against Southampton and Cardiff and were beaten at home by West Brom. If we had won those matches we would be top of the league," he said.
"That is why we are frustrated and upset.
"Of course others haven't taken advantage but I only look at Manchester United.
"It doesn't matter that other teams have dropped points. The most important thing is that Manchester United don't."
That Cardiff result halted the momentum built up immediately before the international break thanks to a win against Arsenal.
Now they must do it again facing a Tottenham side smarting from their hammering at Manchester City last Sunday, which earned Andre Villas-Boas so much scorn and was branded "embarrassing" by the Portuguese coach.
"We are in good form but for that silly goal at Cardiff," said Evra.
"If we had won at Cardiff it would have been a bit easier going to Tottenham. Now we cannot afford to lose."
Moyes now faces a tricky selection dilemma.
Marouane Fellaini will certainly be available after suspension while, prior to the win in Germany, Moyes spoke optimistically about the speed Nemanja Vidic and Robin van Persie will be available for first-team duties after recent fitness issues.
Leaving anyone out after recording a winning margin bettered by United only once in a European away game in their entire history would be harsh, though.
Rio Ferdinand and Ryan Giggs are most likely to drop out on age grounds.
However, the big call would be whether to recall Van Persie should the Dutchman be available.
For all his outstanding form last year, Van Persie has not reached the same level this term and both Wayne Rooney and Shinji Kagawa excelled at the Bay Arena, leading to a school of thought Moyes would be better leaving things as they are for now.