Ms Biden, who was met at the airport by Ms Amer, later had lunch with the Egyptian couple at the capital's Ittihadya palace, located in the leafy eastern suburb of Heliopolis.
She also visited a technical school in Cairo and later met young Egyptians at Al Azhar mosque, the world's premier seat of Sunni Islamic learning.
Al Azhar, located in Cairo's medieval quarter, has long been viewed as a bastion of moderation whose religious discourse disavows extremism and violence.
In keeping with Islamic traditions of modesty, Ms Biden went into Al Azhar wearing a long dress, with her hair covered.
It is the first visit by Ms Biden to Egypt since her husband took office in 2020. Last year, President Joe Biden visited the country for several hours to attend the UN climate summit in the Red Sea resort city of Sharm El Sheikh.
Ms Biden shared a grateful message with Ms Amer after her arrival.
“Thank you for the warm welcome to Egypt, 'Umm Al Dunya', Mrs Entisar Amer!” she wrote on Twitter.
“Umm Al Dunya” is Arabic for “Mother of the World”, one of several nicknames Egyptians and other Arabs sometimes use to refer to Egypt.
Ms Biden's visit coincided with a heatwave in Egypt, with the temperature rising above 40ºC.
Friday's unseasonal scorching heat followed a day when Cairo and other parts of Egypt were hit by a major sandstorm that turned the sky orange and four people died in related accidents.
The US first lady flew in from Amman, where she attended the wedding of Jordan's Crown Prince Hussein and Saudi architect Princess Rajwa. Egypt's first lady also attended the royal wedding.
She will be travelling on from Egypt to Morocco and Portugal, the final stop of her tour.
Her visit to Egypt is focused on the empowerment of women and education for young people, two fields to which Mr El Sisi has been paying much attention since coming to office in 2014.
Ms Biden's spokeswoman Vanessa Valdivia told The Associated Press last week that the first lady's visit to Egypt would also focus on US investments that support educational programmes.
Egypt is one of the world's largest recipients of American economic and military aid. At present, it receives an annual $1.3 billion in military aid.
It is a long-standing US ally but in recent years, the US Congress has sought to condition aid on human rights improvements and reforms.
The close relations between Egypt and the US date back to the 1970s, when in 1979, Egypt became the first Arab state to sign a peace treaty with Israel, Washington's closest Middle East ally.
The two countries also maintain close counter-terrorism co-operation and regularly hold joint war games.