"I believe ... the 21 accords and co-operation protocols that we will sign with our neighbour and friend Greece will mark the beginning of a new era in our relations," he said.
The meeting came as the two nations signed a deal on illegal immigrants, which would allow those coming from Turkey to Greece to be sent back.
"A bilateral agreement was signed to readmit illegal migrants to Turkey," the Greek citizens protection ministry said in a statement.
"The Turkish side has agreed to accept at least 1,000 readmission requests [by Greece] a year," it said.
The issue of illegal migrants has been a major source of discord between the two countries.
Greece and Turkey have been regional rivals for decades, recently divided over issues such as sovereignty in the Aegean Sea and the status of Cyprus.
The two nations came to the brink of war in 1996, and relations were strained again in 2006 when a Greek pilot was killed in a collision with a Turkish plane during a mock dogfight.
Both countries have high military spending, with Greece paying out a higher proportion of gross domestic product on military weapons than any other European Union nation.
Western officials, who have put together a $146bn rescue package for Athens, have pushed for moves adopted by Greece to cut its spending - as high as 5.6 per cent of GDP in recent years - to below three per cent.
"This year we've decided to cut the defence budget from 6.8 billion euros ($8.5bn) to six billion ($7.5bn), that's to say 2.8 per cent of GDP," Panos Beglitis, the Greek deputy defence minister, told French daily Le Monde.
On average, Nato countries each spend around 1.7 per cent of their GDPs on defence.
Barnaby Phillips, Al Jazeera's correspondent in Athens, said: "These two countries are big military spenders, ironically they're Nato allies, and ultimately they would both like their spending to come down".
But Beglitis told Greek radio that arms reductions would come in accordance with a "political condition [with Ankara] that Turkey undertakes specific action and practices in relation to respecting international law on Aegean and east Mediterranean issues".
Erdogan and Papandreou are also expected to chair a joint cabinet meeting with seven Greek ministers on issues including foreign affairs, transport and infrastructure, tourism and culture, education, police and emergency services, energy and the environment.
Papandreou, who helped build stronger ties with Turkey during his stint as foreign minister in a previous government, invited Erdogan in January in a bid to relaunch relations.