He said that Mr Putin’s phone call to him after last month’s failed coup “meant a lot psychologically”.
President Putin received his Turkish counterpart in a Tsarist-era palace outside his home city of St Petersburg.
It was Mr Erdogan’s first foreign trip since last month’s failed military coup, which left Turkey’s relationship with the United States and Europe badly damaged.
The visit is being closely watched in the West, where some fear both men, powerful leaders ill-disposed to dissent, might use their rapprochement to exert pressure on Washington and the European Union and stir tensions within NATO, the military alliance of which Turkey is a member.
Mr Putin said Moscow would gradually phase out sanctions against Ankara, imposed after the Turks shot down a Russian fighter jet near the Syrian border nine months ago, and that bringing ties to their pre-crisis level was the priority.
During a joint news conference after an initial round of talks, Mr Putin said: “Do we want a full-specter restoration of relations? Yes and we will achieve that.
“Life changes quickly.”
Co-operation would be increased on projects including a planned £15.4billion gas pipeline and a nuclear power plant to be built in Turkey by the Russians, Mr Erdogan said, as well as between their two defense sectors.
The Turkish president said: “God willing, with these steps the Moscow-Ankara axis will again be a line of trust and friendship.”
The leaders were to discuss the war in Syria, over which they remain deeply divided, in a subsequent closed-door session.
President Putin received his Turkish counterpart in a Tsarist-era palace
Turkey has been incensed by what it sees as Western concern over its post-coup crackdown but indifference to the bloody putsch itself, in which rogue soldiers bombed parliament and seized bridges with tanks and helicopters.
Some 294 people were killed, many of them civilians.
Mr Erdogan blames Fethullah Gulen, a Muslim cleric who has lived in self-imposed exile in the US state of Pennsylvania since 1999, and his followers for the failed coup.
Turkey has launched a series of mass purges of suspected Mr Gulen’s supporters in its armed forces, other state institutions, universities, schools and the media, prompting Western worries for the stability of the NATO ally.
Denmark’s ruling party said on Tuesday the EU should end accession negotiations with Turkey completely over Mr Erdogan’s “undemocratic initiatives”, the latest European country to condemn developments in Turkey.
Justice Minister Bekir Bozdag said hostility towards the United States was rising among Turks and could be calmed only by the extradition of Mr Gulen, who denies any involvement in the coup and has condemned it.
He said: “There is a serious anti-American feeling in Turkey, and this is turning into hatred.
“It is in the hands of the United States to stop this anti-American feeling leading to hatred.”