The Syrian leader`s trip on Tuesday came three weeks after Russia began air strikes in Syria against Islamic State militants and other forces.
It was Mr Assad`s first overseas trip since civil war broke out in Syria in 2011. The conflict has claimed more than a quarter of a million lives.
On Thursday, a team of Russian MPs is due to meet with President Assad and the head of the Syrian parliament in Damascus.
While in Moscow, Mr Assad made a point of expressing his gratitude for Russia`s military intervention in the conflict.
He said Russia`s involvement had stopped "terrorism" becoming "more widespread and harmful" in Syria.
For his part, Mr Putin said Moscow`s hope was that a "long-term resolution can be achieved on the basis of a political process with the participation of all political forces, ethnic and religious groups".
The BBC`s Steve Rosenberg in Moscow says that by hosting the Syrian leader, President Putin was sending a clear message to the West - that Moscow is a key player in the Middle East, and that there can be no solution to the Syrian conflict without Russia`s involvement.
`Emboldening`
"We view the red carpet welcome for Assad, who has used chemical weapons against his own people, at odds with the stated goal by the Russians for a political transition in Syria," White House spokesman Eric Schultz told reporters.
A state department official said it was not surprised by the visit, but the main US concern was Russia`s continued military support, which he said had emboldened the Assad government - something that would only serve to lengthen the civil war.
In the wake of Mr Assad`s surprise visit, President Putin spoke to a number of Middle Eastern leaders to brief them.
They included the leaders of Saudi Arabia and Turkey, which give support to Syrian rebels.
Mr Putin also spoke to the Egyptian and Jordanian leaders, Russian news agencies said.
Turkish Prime Minister Ahmet Davutoglu said that after the visit "the Syrian government has no legitimacy left".
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