The game appeared to be heading towards a goalless draw until Murphy produced a superb strike in the 80th minute, cutting inside before curling the ball into the far corner.
Fulham responded almost immediately. Substitute Muniz met the ball inside the penalty area and produced a clever finish to beat Lukas Hornicek in the 83rd minute.
Fulham head coach Álvaro Arbeloa was proud of the character his side showed after falling behind.
"We want to be a team that plays in a way that we like to play but, for me, the most important thing is our principle and our character, our mentality, and we have to be a team that always fights until the end, even in the worst condition.
"And we did it pretty well. Obviously, when you are playing at home it's easier because our supporters help us a lot, but I am really proud of the response of the players."
Arbeloa believes his team are making progress in managing matches, adding: "We are a new team, we are progressing. Small steps, but for me as a coach, what I see from my players is they are learning, they are improving, and we are working hard.
"It's going to be a long season, and every week is going to be very, very hard, but I think we are on the right track."
Muniz's impact from the bench was particularly pleasing for the Fulham manager.
"Really important. I felt that the game was for him, because Newcastle were marking us man to man, so we have space in front, and they forced us to play long balls, and I think Rodrigo is very strong in this situation.
"He scored one goal, but they could only stop him with fouls, so I think he did really well. I think it's very important for us, and for me, because I need a player like Rodrigo that can score goals, but he can also create so many chances for us."
Newcastle United manager Matthias Jaissle, meanwhile, provided an encouraging update on teenager Sean Steur, who was withdrawn after an awkward fall.
"I think he is good, he's fine. But he felt the physicality which is not 100% there yet. It's something I mentioned already before. This is normal if you sign a young lad. He needs to get used to it, but he did an amazing job."
Jaissle also felt his side had improved after their previous defeat.
"Yes. I believe we did enough to win the game. I think, compared to the last match, we spoke really during the week, and we saw that from people also. There was a complete commitment on the pitch but we were better in all phases of the game. Especially as the game wore on.
"The first 20 minutes, we saw the lack of confidence from last week's performance but the way that we grew then throughout that game was excellent."