Mr Erdogan, who was re-elected last month, vowed to “propel our country forward” with his sweeping new powers.
But his opponents fear that his new role as executive president will destroy Turkish democracy.
Mr Erdogan’s new position marks a transition away from a parliamentary system and the office of prime minister, which has been in place since the foundation of the modern Turkish republic 95 years ago.
It allows him to appoint ministers and vice-presidents and intervene in the legal system.
After taking the oath of office in parliament on Monday, Mr Erdogan told guests at the presidential palace in the capital, Ankara, that Turkey was “making a new start”.
“We are leaving behind the system that has in the past cost our country a heavy price in political and economic chaos,” he said.
Following news of the appointment of Mr Erdogan’s son-in-law, the Turkish lira lost more than 2% of its value.
It also emerged that Mehmet Simsek, a former banker at Merrill Lynch who acted as deputy prime minister in Turkey’s previous government, will not hold a position in the new cabinet.
In another key change announced by Mr Erdogan on Monday, military chief Gen Hulusi Akar was named as the new defence minister.
Meanwhile Mevlut Cavusoglu remains in his post as foreign minister.
Last month Mr Erdogan was re-elected with 53% of the vote. He has presided over a strong economy and built up a solid support base.
But he has also polarised opinion, cracking down on opponents and putting some 160,000 people in jail.