United central defender Aaron Downes crashed home an 82nd-minute opener from Kevin Nicholson's inswinging corner from the left.
Argyle's luck looked to be out when United keeper Michael Poke went full stretch to keep out Robbie Williams' rising drive, but the industrious Harvey had other ideas.
When the ball spilled to the teenage striker on the edge of the box he drilled it past the diving Poke to send the Green Army in a 10,003 crowd into raptures.
Fletcher said: "He did well got his goal. I am pleased for him, obviously he is a local lad and he's come in and shown all the exuberance and energy and all of those sort of things you would expect from a young lad making his debut.
"He showed a bit of quality at the end there with his goal which was pleasing.
"They were terrible conditions, it was windy and wet, I thought we played really well.
"We pretty much dominated the game so to then fall behind was pretty disappointing, but I thought our reaction once we went a goal behind was good."