Chris Wood thought he had given the home team the lead in the 25th minute, but he was adjudged to have been offside after Nikola Milenkovic knocked down a header to the striker after a delivery from Ola Aina into the box.
Wood's disallowed goal was the biggest talking point of the first half, with neither side creating much in the opening 45 minutes.
However, Fulham went on to take the lead thanks to Raul Jimenez in the 51st minute when he converted a penalty which was awarded after VAR deemed there was a clear foul on Andreas Pereira by Murillo and recommended an on-field review.
After the penalty, Nuno's side had six of the game's next 12 shots, but they rarely threatened Bernd Leno's goal.
The visitors dominated possession throughout and were able to prevent the hosts from creating a single big chance.
Fulham managed to hold on for all three points, and the result leaves them with 11 points from six games, and they have now won three of their last five league games.
As for Nottingham Forest, they stay on nine points and ended the day in ninth place, two points ahead of the likes of Manchester United and Tottenham Hotspur.
NOTTINGHAM FOREST VS. FULHAM HIGHLIGHTS
Raul Jimenez goal vs. Nottingham Forest (51st min, Nottingham Forest 0-1 Fulham)
Raul Jimenez scores from the penalty spot after the referee reviews a foul from Murillo on Andreas Pereira, and the striker stutters as he gets closer to the ball, before shooting into the right side of the net.
Jimenez has scored in his last three Premier League games for Fulham!
MAN OF THE MATCH - JOACHIM ANDERSEN
Joachim Andersen was vital for Fulham from a defensive perspective, and the former Crystal Palace centre-back made 10 clearances, two blocks and one interception.
Andersen was also vital for Silva when Fulham had the ball, with the defender completing 75 of his 84 passes, the most of any player on the pitch.
NOTTINGHAM FOREST VS. FULHAM MATCH STATS
Possession: Nottingham Forest 47%-53% Fulham
Shots: Nottingham Forest 6-7 Fulham
Shots on target: Nottingham Forest 1-1 Fulham
Corners: Nottingham Forest 3-1 Fulham
Fouls: Nottingham Forest 4-5 Fulham
BEST STATS
WHAT NEXT?
Nottingham Forest travel to face Chelsea in their next Premier League game on October 6, and will have to hope that Cole Palmer does not replicate his showing against Brighton and Hove Albion, a match in which he scored four times.
As for Fulham, they will face champions Manchester City at the Etihad on October 5, and will look to take advantage of the absence of Rodri from City's midfield.