Joey Barton's organised outfit were a tough nut to crack for long spells but Sabri Lamouchi's side earned a place in the second round as Silva controlled, turned and finished wonderfully from 10 yards to provide the only goal of the game.
Forest made nine changes to the team that claimed a 1-1 draw at Leeds, while giving a first start to seven of their 12 summer signings.
It was one of those, Carl Jenkinson, who almost gave them an early lead with a dipping volley that was smartly saved.
Fleetwood, in contrast, made only two changes to the team that had beaten AFC Wimbledon 2-1 in League One.
Sammy Ameobi flicked a header narrowly over as Forest continued to try to break Fleetwood's resolve after the break.
The goal finally came in the 59th minute as John Bostock slid in Silva, who lifted an emphatic finish high into the back of the net.
Ross Wallace was denied a spectacular equaliser when Samba saved his curling free-kick amid a late flurry from the visitors.