The midfielder fired a wonderful long-range effort into the top corner as the League One side made life difficult for their top-flight opponents, while Joe Edwards saw a header cleared off the line as they searched for an equaliser before Angel Di Maria added a second late on.
United fired several long-range efforts off target while James Wilson fired a low shot straight at Jed Steer as the visitors failed to create too many clear-cut chances.
But Gary Johnson’s men couldn’t take advantage of a lacklustre first-half display from the visitors, and Louis van Gaal earned a slender win in his first-ever FA Cup tie.
The Manchester United boss made several changes but named a strong line-up for the visit to Huish Park, with Wayne Rooney and Radamel Falcao leading the line for the visitors.
The first half was a scrappy, tense affair as the Glovers worked hard to close the visitors down and not allow them any time on the ball.
It was a tactic that worked as United failed to register a shot on target until the 32nd minute, when James Wilson drilled a low shot at the goalkeeper after a good ball through by Ander Herrera.
Kieffer Moore and Paddy McNair ended up with matching bandages after a nasty clash of heads which resulted in a lengthy stoppage as the two were patched up.
Rooney saw one effort fly over the bar while several other long-range efforts were blocked in a disappointing opening period from Van Gaal’s men.
The first big chance of the match fell to the hosts as Smith’s ball into the box was helped on by Evans, and Moore latched on to it but couldn’t get it out of his feet as he scuffed a low effort straight at David De Gea.
But it was the visitors who struck the decisive blow with 26 minutes remaining, as Herrera latched on to Fletcher’s deflected pass before spinning and firing a stunning 25-yard effort into the top corner.
Van Gaal’s men relaxed after breaking the deadlock but the Glovers had a superb chance to equalise, with substitute Di Maria clearing Joe Edwards’ far-post header off the line.
And the visitors confirmed their place in the fourth round in the 90th minutes as Rooney fired a superb ball beyond the defence for Di Maria, who raced into the box before chipping the ball past Steer to seal a 2-0 victory.