The leaders discussed the crisis in Nagorno-Karabakh and were due to officially open the construction of a natural gas pipeline linking the two countries, the Turkish president’s office said. The pipeline is expected to help diversify Nakhchivan's gas imports away from Iran.
Mr Erdogan hailed the defeat of separatist forces and said he hoped Armenia would take "sincere steps" to normalise relations and ensure regional stability.
"The window of opportunity has opened to settle the situation in the region. This opportunity must not be missed," Mr Erdogan said, adding that Baku's success in the military operation was a source of pride for Turkey.
"We expect from the Armenian side that they will shake the hand stretched out to them in peace and will take sincere steps."
The Turkish leader also said he was pleased to "connect Nakhchivan with the Turkish world” as he signed a deal for the gas pipeline.
During the meeting, Mr Aliyev promised that the rights of ethnic Armenians in the Nagorno-Karabakh would be protected after Baku's victory.
Aid being sent from Baku "shows once again that Karabakh's residents, regardless of their ethnicity, are citizens of Azerbaijan. Their rights will be guaranteed by the Azerbaijani state", he added.
At least 4,850 people had crossed into Armenia by late Monday morning, the Armenian government said in a statement, up from about 3,000 people seven hours earlier.
Armenian media showed buses of displaced people arriving from the disputed region, many without belongings.
The political fallout continued as Moscow accused Armenia of trying to sever bilateral ties after Yerevan accused Russian peacekeepers of failing to stop the Azerbaijani military offensive.
"The leadership in Yerevan is making a huge mistake by deliberately trying to destroy Armenia's multifaceted and centuries-old ties with Russia, and by holding the country hostage to the geopolitical games of the West," Russia's foreign ministry said.
Washington expressed its alarm at the Karabakh crisis as US Agency for International Development (USAID) chief Samantha Power and US State Department Acting Assistant Secretary for Europe and Eurasian Affairs Yuri Kim arrived in Yerevan.
"The United States is deeply concerned about reports on the humanitarian conditions in Nagorno-Karabakh and calls for unimpeded access for international humanitarian organisations and commercial traffic," USAID said in the announcement of the trip.
Protesters have been demonstrating against the conflict in Yerevan, calling for the ousting of Prime Minister Nikol Pashinyan over perceived failure to support Armenian separatists in Nagorno-Karabakh.
Armenian police detained at least 142 protesters, Russia's Tass news agency said.
Moscow has 2,000 peacekeeping troops in the region, but Armenia says it is not doing enough to protect civilians.
Turkey is looking to use the conflict to seek more concessions from Armenia and win Azeri control over a strip of land along Armenia’s border with Iran, Bloomberg reported.
The six-week war was the second fought over the territory in 30 years.
Armenia says more than 200 people were killed and 400 wounded in last week's Azeri operation, a hostility condemned by the United States and other western allies of Armenia.
Baku launched the operation to disarm what it has called terrorists from the region.
On Sunday, Mr Pashinyan warned up to 120,000 Armenians could flee Nagorno-Karabakh amid fears of ethnic cleansing.
“If proper conditions are not created for the Armenians of Nagorno-Karabakh to live in their homes and there are no effective protection mechanisms against ethnic cleansing, the likelihood is rising that the Armenians of Nagorno-Karabakh will see exile from their homeland as the only way to save their lives and identity,” he said.
David Babayan, an adviser to Samvel Shahramanyan, president of the self-styled Republic of Artsakh, told Reuters, that "99.9 per cent prefer to leave our historic lands".
Speaking at the UN General Assembly last week, Mr Erdogan said he hoped a "comprehensive peace agreement" could be signed soon, but repeated his support for Azerbaijan.
“Everybody has the right to coexist on the Azerbaijan soil, including the Armenians and that should be our primary goal," he said.
"We are moving together with Azerbaijan under the motto that we are two nations, one state.”