Israel's PM-elect and Mr Erdogan held a 12-minute phone call and pledged a "new era" for bilateral relations, said a spokesperson for Mr Netanyahu's Likud party. Mr Netanyahu also expressed condolences over the deadly Istanbul bombing, which killed six people last week.
Mr Erdogan told Mr Netanyahu relations must be maintained on "mutual understanding", the Turkish presidency said.
Turkey and Israel have taken numerous steps to repair relations in recent months following years of estrangement. Defence Minister Benny Gantz visited Ankara in October to restore security ties, while President Isaac Herzog met Mr Erdogan in March during the first Israeli state visit to Turkey in 14 years.
Turkish Foreign Minister Mevlut Cavusoglu travelled to Israel in May, where he said ties would be boosted after "frank" talks over Jerusalem. Outgoing Israeli Prime Minister Yair Lapid visited Ankara the following month.
On Friday, Ankara appointed an ambassador to Israel after a four-year vacancy. Irit Lillian was announced as Israel's envoy to Turkey in September.
Ties between the two countries broke down over Mr Erdogan's staunch support for the Palestinian cause, and hit rock bottom when 10 Turkish citizens were killed in a 2010 Israeli raid on an aid ship heading to Gaza. Israel has also voiced concern over Turkey's hosting of senior Hamas operatives.
Diplomatic relations were restored in 2016, but two years later Turkey recalled its diplomats from Israel and expelled Israeli envoys when Israeli forces killed several Palestinians protesting in the blockaded enclave.
Turkey helped to thwart an Iranian plot to target Israeli tourists visiting the country in June, and Mr Netanyahu offered Ankara more security assistance during Thursday's call with Mr Erdogan.
The Turkish President is said to have offered condolences to Mr Netanyahu over the death of three Israelis in a Palestinian attack on the West Bank settlement of Ariel on Tuesday.
Mr Erdogan congratulated Mr Netanyahu last week after his victory in Israel's fifth election in less than four years put him on course for a sixth term as prime minister. The Turkish President said he hoped for co-operation "that will bring peace and stability to our region".
He had said previously that Turkey wished to maintain ties with Israel regardless of the election outcome.
"Whatever the election result, we want to maintain relations with Israel on a sustainable basis, based on mutual respect for sensitivities and common interests," Mr Erdogan told Turkish broadcaster ATV after the election, which set the stage for Israel to welcome its most right-wing government in history.