Brighton took the lead after just 12 minutes through Brajan Gruda before ex-Man United striker doubled their lead in the 64th.
Benjamin Sesko managed to pull one back late on, but it turned out to be nothing more than a consolation.
Their loss to Grimsby earlier in the season means before today, the last time Man United were knocked out at the first hurdle in both domestic cups was 1981/82.
Speaking after the game, Fletcher admitted that he believes that Man United players are ‘fragile.’
"Mixed performance. We started OK and then the goal knocked the stuffing out of us,” he said.
“I challenged them at half-time to move the ball with tempo instead of just composure. When we got back to 2-1 I thought here we go, the crowd was up, the energy was up but we couldn't get it over the line and get an equalising goal.
"I think you could see the players are fragile but they've got to respond. Confidence is one of the most powerful things in football so when you don't have it, you have to dig deep and then the confidence comes back.
"It's over to them, they need to make sure that they have got a lot to play for this season. This team is still good enough to achieve success this season but they have to dig deep."