The first of a strong British contingent to reach a semi-final, where each of the losers is awarded a bronze, Buatsi pounded Algeria's Abdelhafid Benchabla for a 3-0 points decision.
"I don't go looking for the knockout, I just want to dominate in every way that I can," said the Ghanaian-born 23-year-old, who handed out a standing eight count in each of the first two rounds but then took some punishment in the third.
He said he had wanted to show that he could take the punches as well.
"You can give and take, entertain a little bit and we go back to the boxing. I'm glad to have put on a bit of a show and secure bronze at least," he added.
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Mayweather, who took a bronze medal for the United States at the 1996 Atlanta Games as a featherweight before turning professional, met the Briton on Saturday and was back to watch some of the earlier fights on Sunday.
The former number one, who retired with a 49-0 record last September after winning world titles in five weight divisions and with earnings in excess of $800 million, said on social media he was in Rio "scouting future boxing champions".
"He just said 'you're a great fighter, I've heard about you before I got here and you are a future world champion'," said Buatsi.
"It was good to hear that from someone like Mayweather. He's the greatest boxer of our generation that I have seen personally so it was good to hear that from the champion."
The Briton's next fight for a shot at gold will be Kazakhstan's Adilbek Niyazymbetov, the second seed in the draw and the 2012 silver medallist.
Cuba's Julio Cesar La Cruz and France's Mathieu Bauderlique will contest the other semi-final.
While Buatsi went through, there was disappointment for light-welterweight team mate Pat McCormack who lost 2-1 to Yasnier Toledo of Cuba in a last 16 bout.
"I got a terrible draw, the Kazakh first and then the Cuban," said McCormack. "I think I started too slow, I should have put it on him from the first bell.
"I think he's the best in the world ... I know I'm mixing with the best and I will be one of the best in a couple of years when I've got a bit more experience."
(Editing by Greg Stutchbury)
