Pompey boss Michael Appleton was forced to field nine teenagers in his starting line-up and they held their hosts until Johnny Gorman, Paris Cowan-Hall and Nick Chadwick fired Fletcher's side into the second round.
Fletcher said: "I am pleased to get through because it was a potential banana skin for us.
"I think a lot of credit has got to go to the Portsmouth players. They put real pride back into the shirt and did so well for their football club in really tough circumstances.
"It was not so long ago that we were going through the same sort of thing and you feel for everyone at the club and hopefully they can get through it sooner rather than later."