Borja Mayoral put Getafe ahead in the 11th minute at the Santiago Bernabeu before Joselu levelled after half-time and Bellingham, 20, scored a dramatic late winner, his fifth goal in four appearances.
The result means Madrid have taken maximum points this season and top the LaLiga table with 12 points.
"It's the loudest moment I've heard in a stadium," Bellingham told Real Madrid TV. "I can't believe it to be honest. When [the crowd] were singing 'Hey Jude' at the end, I got goosebumps. I just wanted to turn and stand still and listen to it.
"My legs were shaking," Bellingham said. "I know that I was brought in for these kind of moments. They're the kind of moments I know I can deliver for this team. I want to keep doing it and keep making sure the fans go home happy."
"He's a very serious professional," coach Carlo Ancelotti said when asked about Bellingham's adaptation. "It can't be any other way, the professionalism in this squad is very high. He has a lot of role models. He's adapted very well."
Ancelotti backed Bellingham to keep scoring.
"He can get to 15 [goals] without any problems," Ancelotti said. "He did it last year with Dortmund. He stands out because he moves very well without the ball. That's his best quality."
"He's a player who learns very fast," Ancelotti added. "You don't have to tell him things many times... We've shown him videos of when he played at Dortmund, of things we liked.
"What's changed is that he moves a lot more without the ball into the opposition box. At Dortmund he used to play between the lines."
United let the forward leave following a six-month investigation into his conduct, after a criminal case over allegations of attempted rape and assault was discontinued.
"[Greenwood's] is a delicate situation," Getafe coach Jose Bordalas said. "Everyone knows what happened, and everyone knows how it ended, without a guilty verdict... We all know the potential he has. He wants to recover his professional status, and Getafe can help him in that sense."