The Red Devils need to avoid doing any more serious damage to their Premier League title challenge.
Having spent so many seasons dishing out last-gasp defeats to their rivals, Sir Alex Ferguson's side received a taste of their own medicine as Everton fought back from 3-1 down to snatch a draw with two goals in injury time at Goodison Park.
In their only other away game so far this season United saw Fulham's Brede Hangeland earn a 2-2 draw with an 89th-minute header - moments after Nani had missed a penalty.
Those four points are the difference between the third-placed Red Devils and the leaders and defending champion Chelsea, who are the only side with a 100% record.
It may only be early days but Fletcher knows allowing winning positions to slip from their grasp will become costly.
"We should have learned from the Fulham game and we didn't," said the Scotland international, who got his side back into the game with a close-range volley just before half-time to cancel out Steven Pienaar's opener.
"We really have to learn from this game if we want to take it forward this season and win the league.
"We can't afford to throw away points as we have done in the two away games so far.
"It just shows you that you think you are winning 3-1 and can take it easy but you've got to keep taking your chances and really finish the game off.
"These are moments you can look back on and say 'That should have been three points' but it is a long season and there are many twists and turns to come."
Fletcher was keen to stress that the recent late lapses should not be viewed as a sign of weakness.
However, he admits if United continue to give teams hope when there should be none then they will find themselves under pressure more regularly.
"It just shows you that you have to play right to the final whistle in this league because teams don't give up, especially when they are at home," added the 26-year-old.
"The more confidence you give to teams, showing them things like that, the more confidence they will get to say 'Keep on right till the end against Manchester United'.
"We need to put that to bed and get away from it as soon as possible.
"There is no vulnerability, it is just simple mistakes which we need to eradicate quickly."
Without Wayne Rooney, who was left out of the squad entirely after a week of lurid allegations about his private life, United began poorly at Goodison.
Everton should have been well ahead before Patrice Evra's overly-ambitious but failed overhead-kick clearance allowed Mikel Arteta to race through on goal and have Pienaar convert his rebounded shot.
United clicked into gear just before the interval when Fletcher nipped in at the near post to volley home Nani's cross and within three minutes of the second half kicking off Nemanja Vidic had headed them in front.
When Dimitar Berbatov fired home with the outside of his right boot with 24 minutes to go the match looked over until chaos broke out in the visitors' penalty area in added time with Tim Cahill powering in a header before Arteta's deflected strike earned a point.
"The game was fizzling out but once you give a team like Everton a bit of hope with the second goal - which came out of nothing - it lifted the whole place," said Fletcher.
"They found that extra bit of energy and threw men into the box in the last minute.
"You can't afford to give teams at their home ground any sort of momentum or confidence, a half-chance or a goal, because in the end you suffer for it.
"The manager was really disappointed and made it clear in no uncertain terms of what is expected at this club."