Abughaush beat Alexey Denisenko of Russia 10-6 to win the men’s under-68kg division while Alizadeh landed bronze in the women’s under-57kg category by beating Nikita Glasnovic of Sweden 5-1.
Abughaush took a congratulatory phone call from the Jordanian royal family after his win and dedicated victory to the kingdom.
“It’s an indescribable feeling to win the first medal in the history of Jordan in all the sports,” he said.
“It’s also a great feeling to listen to the national anthem of Jordan being played in Rio in front of the whole world.”
The women’s under-57kg weight division saw Briton Jade Jones retain her Olympic crown, easily despatching Eva Calvo of Spain 16-7 in the final.
“I am so happy for Iranian girls because it is the first medal and I hope at the next Olympics we will get a gold,” said Alizadeh.
“I am very excited and I want to thank my parents and my coach. They really stand behind me and I am so happy.”
Great things have been expected of Alizadeh, who won the Youth Olympics two years ago and last year took bronze at the world championships, beating Jones in the quarter-finals.
But the Briton came up trumps in Rio.
“It feels unbelievable. I didn’t realise how much pressure I would feel but to pull this off is amazing,” Jones, 23, told the BBC.
“I am still young so to be double Olympic champion already is crazy. I’ve been crying in between sessions because I was so stressed but I did it when it mattered. I’m buzzing.”
Yet her achievement paled in comparison with that of Alizadeh, who has had to overcome innumerable obstacles in her home country.
Deeply conservative Iran has had few women competitors at the Olympics, particularly since the Islamic revolution in 1979.
Archer Lida Fariman was the first to be allowed to compete at the 1992 Games in Barcelona.
But clothing obligations imposed by hardline religious leaders continue to place barriers to women competing freely, and on a level pegging, in many sports.
Women are also banned from watching sports in Iran where men are present.
Earlier in these Games, an Iranian woman was asked to leave the volleyball arena in Rio after she held up a banner protesting against her country’s laws.
Darya Safai held a sign saying: “Let Iranian women enter their stadiums.”
Safai, who was born in Iran but lives in Belgium, was allowed to stay after refusing to leave the Iran-Egypt men’s match.
South Korea’s Lee Dae-hoon and Joel Gonzalez of Spain, a gold medallist in London, won the men’s under-68kg bronze medals, while Hedaya Malak of Egypt landed the other women’s bronze.