Mr Putin called the visit by Mr Tsipras “very timely” and hailed the two countries’ “common spiritual roots”.
Greek officials have previously pointed to Russia as a possible alternative source of financial assistance to alleviate the country’s debt crisis.
But analysts say Russia’s own economic woes mean any help would be limited.
Greece’s new government is embroiled in negotiations with the EU and IMF to unblock a bailout package and could run out of funds within weeks.
Russia was among Greece’s leading trade partners before sanctions on its energy industry and Greece’s own economic woes dropped trade between the two countries by 40%.
Mr Tsipras received a warm welcome in Moscow, which has seen its ties with the EU strained over Russia’s actions in Ukraine.
The Russian president said the two leaders would “analyse what the two of us could do to restore the previous rate of growth” in trade.
Mr Tsipras said he aimed to “restart relations for the good of both our people”.
Russia is not in a position to solve Greece’s economic issues however, said Constantinos Filis from the Institute of International Relations.
“Russia is not and cannot be a (EU) substitute for Greece. It can only be a supplementary option,” he said.
Mr Putin and Mr Tsipras are also expected to discuss ties between the EU and Russia in the wake of the Ukraine crisis.
Before his arrival, Mr Tsipras described the sanctions imposed by the EU and US on Russia in the wake of its annexation of the Crimea as “a road to nowhere”.
The European Parliament President, Martin Schulz, said Mr Tsipras should not break with the EU line on sanctions.
“Greece demands and gets a lot of solidarity from the EU. We can therefore also ask for solidarity from Greece and for this solidarity not to be ended unilaterally by pulling out of joint measures,” he told a regional German newspaper, the Muenchner Merkur.
Russia imposed a ban on many western food imports in retaliation, but Agriculture Minister Nikolai Fyodorov has said the government could consider removing three countries, including Greece, from the embargo, Russian state media reported.
Mr Tsipras came to power pledging to end austerity, but his plans have met resistance from Greece’s EU/IMF creditors, who lent the country billions to help it avoid bankruptcy.
Greece has not received bailout funds since August last year, with the EU and IMF dissatisfied with the pace of Greek reforms.
A Greek repayment of €448m to the IMF is due this Thursday.
On Tuesday, the Greek government said Germany owed Greece nearly €279bn (£204bn; $303bn) in war reparations for the Nazi occupation during World War Two.
It is the first time Greece has calculated what Germany allegedly owes.
But Germany says the matter was resolved legally years ago, and reacting to the claim Economy Minister Sigmar Gabriel said it was “dumb” to link Greece’s bailout with the question of war reparations.