Evra scored the second goal in United's 2-1 win over Arsene Wenger's side, and with Chelsea and Manchester City drawing at Swansea and West Ham, respectively, Sir Alex Ferguson's team now lead the table by one point.
But despite picking up maximum points from games against two of their closest rivals, France international Evra says there is room for improvement after his side wasted numerous chances on Saturday.
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He continued: "The expectations are very high here and we should have scored more goals - we felt like we had lost the game in the dressing room afterwards."
Despite only winning by one goal in the end, Evra suggested it is important for his team-mates to remember the calibre of opposition they defeated: "If we had taken every chance we created it would have been a different game, but to only finish 2-1 is a bit disappointing.
"When [your expectations] are so high you can sometimes forget that you've beaten a team like Arsenal. It's a big game we have won," he added.
And the 31-year-old insists United can now look to reclaim the title they lost to Manchester City if they continue their form against the league's strongest sides: "We've won at Liverpool, at Chelsea and now we have beaten Arsenal. Last season we didn't do that and that's why as a team we know we have the potential and the quality to win the league."
Evra's second goal of the season gave United a two-goal cushion - one they needed considering Santi Cazorla's late consolation for Arsenal - but the left-back insists points are his only concern: "Accidents can happen and it's happened twice now this season! I'm really happy to score again but the most important thing is the three points."