The TCG Bayraktar, which set sail from Beirut around 8:00 am (0500 GMT), pulled into the port of Mersin around 9:00 pm carrying the first batch of 966 people.
As it pulled into the port, a large group of journalists were waiting on the quayside.
A second boat carrying a similar number of passengers was set to dock around midnight, officials said.
Turkish officials said around 2,000 people had signed up to leave Lebanon on navy ships in the first such evacuation by sea.
Turkey, which is estimated to have 14,000 citizens registered with its consulate in Lebanon, announced the move on Tuesday because of the deteriorating security situation in the country.
Speaking to AFP in Beirut ahead of the boats' departure on Wednesday, Turkish ambassador in Lebanon Ali Baris Ulusoy told AFP the evacuation was a first and came after "a growing number of requests from our Turkish nationals living in Lebanon to be evacuated".
He said they were allowing close relatives "who are not necessarily Turkish nationals, like spouses, children or parents" to travel too.
Israel and Lebanese militant group Hezbollah have exchanged near-daily fire on the country's southern border since the start of the Gaza war a year ago.
But Israel has intensified its strikes against Hezbollah since September 23, killing more than 1,200 people in Lebanon and displacing more than a million from their homes.