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A wonder goal from Ander Herrera and trademark Angel Di Maria chip at Huish Park on Sunday earned United a place in the draw.
Louis van Gaal's side were far from their best, but Fletcher is hopeful the triumph could be the start of a golden cup run for a team who have not lifted this trophy since 2004.
"We knew it wasn't going to be easy," he told BT Sport. "You have to battle before you can win the match.
"We battled and then Ander scored a fantastic goal and sometimes that is what it takes.
"We're in the draw for the next round and it's been too long since Manchester United won this competition."
Herrera was pleased to come out on top, yet conceded United were not at their best against League One's bottom side.
He added: "We knew that it was going to be a very difficult game. They played very good and we had to work hard.
"We are very happy, but I think we could have played better before the first goal."
