Stavros Flatley won the public vote with their Zorba the Greek dance routine, with father Demitri telling his son Lago that he had the time of his life dancing with him, no matter what the result.
Jafargholi beat the MD Show Group in the judges' vote-off with his take on 'And I Am Telling You I Am Not Going' from the Dreamgirls musical.
Judge Piers Morgan described the song as "the best singing performance of the semi-finals".
Fellow panellist Simon Cowell said: "I'm not sure if it was as good as the first performance, but I'm absolutely certain we're going to see you on Saturday, and like I said to someone yesterday, you're a real contender now to win this competition."
Unsuccessful semi-finalists on last night's show included Harmony, Floral High Notes, The Barrow Boys and Kay Oresanya's human saxophone.
Jafargholi and Stavros Flatley join singers Shaun Smith and Susan Boyle in the final, as well as dance troupes Flawless and Diversity.
The fourth Britain's Got Talent semi-final takes place tomorrow on ITV1 at 8.30pm.