The 18-year-old, who took the stage during the Miami auditions, was supported by family members backstage who cheered her on and urged her to "use [her] gift".
Amaro said in voiceover: "I don't want to disappoint any of my family. They're all so proud and I've gotta definitely get that 'yes.'"
When she was asked by Cowell if she was "any good", she replied: "I'll let you determine that."
LA Reid - who was seen nodding his head in time to Amaro's performance - mouthed the word "wow" as the song came to a close.
"Melanie, Melanie, you are truly amazing. You have the gift, you have the soul, you have the swagger, you have the spirit... you have the X Factor," he commented.
A tearful Nicole Scherzinger told Amaro: "This is why I do this. People like you inspire me," while Paula Abdul added: "You have a unique ability unlike anything we've heard. I don't think it's like anything I've ever heard in any audition that I've been in."
Simon Cowell said: "When I was asked by a lot of people why I was bringing this show to America, I hoped it was because we were going to find someone like you."
Amaro received yeses from all four judges, having wowed them with her larger-than-life vocals and earning a standing ovation from the audience and panel alike.
During the vote, Reid added: "Everything I've walked away from to do X Factor, you've just made it all worthwhile. Absolutely, yes."
Scherzinger commented: "You made a believer out of me. Thank you. Yes."
Abdul said: "You are the brightest gift, God bless you. Yes," while Cowell played to the audience, declaring: "Miami, help me here."
As they cheered, he added: "It's four yeses. Well done."