The two leaders met on Tuesday to discuss "regional and international topics of mutual importance".
Ankara and Khartoum said the island deal would increase tourism and create a transit point for pilgrims crossing the Red Sea to Islam's holiest city, Mecca.
Egyptian and Saudi media slammed the agreement and alleged that Turkey would build a military base on Suakin island harming the security of Arab countries - a claim denied by Sudan and Turkey.
Ankara strongly condemned Egypt's military coup in 2013, which overthrew democratically elected President Mohamed Morsi of the Muslim Brotherhood.
Cairo responded to the island deal by sending troops to Eritrea, which has borders with both Sudan and Ethiopia. Khartoum then shut its border with Eritrea. Sudan also recalled its ambassador to Cairo.
Ethiopia, which already has thousands of troops on its border with Eritrea, responded by sending more troops to its border with Eritrea.
Tensions over Nile River water, dam
Political tension between the countries has been rising for years over the use of the water of the Nile River and Ethiopia's decision to build the continent's biggest hydroelectric dam on the river,
Egypt has been at odds with Sudan and Ethiopia over the $4.8bn dam project, with Cairo fearing that its position downstream may affect its access to water from the Nile River basin, which will feed the dam.
Cairo accuses Sudan of supporting the Grand Ethiopian Renaissance Dam (GERD) project, while Khartoum accuses Egypt of supporting rebels in Sudan.
Meanwhile, Eritrea has fought two border wars with Ethiopia - which has had a decades-long dispute with Egypt over the Nile River water. The border wars left more than 80,000 people dead and the two East African countries are technically still at war.
Afwerki's visit to Egypt comes days after he visited the United Arab Emirates, an ally of Egypt, to discuss military cooperation between the two countries.
The UAE has military bases in Eritrea and Somalia, which also has a long border with Ethiopia.