United overcame League One's bottom club 2-0 at Huish Park thanks to goals from Ander Herrera and Angel di Maria.
Scotland midfielder Fletcher - who played the full 90 minutes in Somerset - is one of just three members of the squad to have won an FA Cup winners' medal, though he is the only one to do so with United having featured in the 2004 triumph over Millwall.
United have since reached the final in 2005 and 2007 but came up short on both occasions.
And Fletcher believes the time has come for Louis van Gaal's men to win the competition for a 12th time.
"We knew it wasn't going to be easy. It was going to be difficult with tackles flying in," Fletcher said.
"You have to battle before you win the match and we did that. We battled for 60, 70 minutes and then Ander scored a spectacular goal and sometimes that's what it takes.
"The longer you can keep possession, although it was difficult, the more you tire the opposition out and you can win the game in the last 20 minutes.
"We knew it was going to be difficult but we're in the draw.
"It's been too long since United have won this competition.
"We really want to win it this year and, hopefully, we'll look back on our result here as one of the most important ones if we go far in the competition."
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
