President Nicos Anastasiades and Turkish Cypriot leader Mustafa Akinci met in Nicosia's neutral buffer zone.
UN envoy Espen Barth Eide said they "shared their vision for a united Cyprus".
Mr Akinci was elected last month. The Turkish-controlled north broke away in 1974 after a brief Greek-inspired coup.
Since a Turkish military invasion in 1974 the island has been effectively partitioned, with the northern third inhabited by Turkish Cypriots and the southern two-thirds by Greek Cypriots.
Peace negotiations stalled last October, when the Greek Cypriots walked out in protest at the presence of a Turkish ship prospecting for natural gas off the island's south coast.
Correspondents say there are hopes that Mr Akinci, a leftist moderate, can push forward the new round of negotiations.
"The talks took place in a very positive climate and I believe that working in similar fashion, we can hope for progress," Associated Press quoted Mr Anastasiades as saying after Friday's UN-brokered meeting.
In an initial move, Turkish Cypriot authorities agreed to no longer require Greek Cypriots to fill in visa forms at crossing points to northern Cyprus, UN envoy Mr Eide told reporters.
Compensation for those displaced by the conflict is expected to be one of the thorniest issues in the talks.
UN peacekeeping forces estimate that 165,000 Greek Cypriots fled or were expelled from the north, and 45,000 Turkish Cypriots from the south, although the parties to the conflict say the figures are higher.
The self-declared Turkish Republic of Northern Cyprus is diplomatically isolated, recognised only by Turkey.

Marchant, 20, has made 37 appearances for Quins since making his senior debut in May 2015, scoring 11 tries.
"I'm delighted to have re-signed with Harlequins," he said. "This is such a great club and I have loved my time here. It was an easy decision to stay."
Marchant had been due to tour Argentina with England last month but was ruled out with a toe injury.
The leaders of the divided Greek and Turkish communities in Cyprus have resumed peace talks amid a "climate of optimism", the UN mediator says.
[NEXT_CONCEPT]
Harlequins centre Joe Marchant has signed a new and extended contract with the Premiership club.