Putin delivered an unscheduled TV address after the strike, in which he told Russians that the Ukraine conflict had taken on "elements of a global character".
He also threatened more strikes with new weapons on Ukraine and did not rule out strikes on the West.
"Russia will overcome everything... Nobody can defeat it," said Alexei Peshcherkin, a 57-year-old plumber.
He said Putin was "doing everything very well", adding: "He does not give a chance to anyone threatening Russia."
Russia said its launch of a nuclear-capable missile was a response to the first strikes by Ukraine with US- and UK-supplied long-range missiles at Russian territory in recent days.
Putin supporters welcomed the hawkish rhetoric.
Alexander Timofeyev, a 72-year-old railway company employee, said the speech "made me feel secure".
"There are no questions if you have someone and something protecting you," he said.
"They're not stupid, the people who are trying to intimidate us" in the West, he added.
"But they are afraid. And fear is good sometimes."
Timofeyev said he believed a Third World War was "not very likely".
- 'Russia will overcome' -
Yulia Kim, a 52-year-old doctor, said there was an "escalation" underway between Moscow and the West.
"I am worried that a nuclear war will start," she said, as she walked through a central Moscow square.
"But we have to fight for our independence and resist to the end."
Andrei, a 61-year-old economist, said he was sure "Russia has sufficient means to defend its independence".
"Russia will overcome everything, together with its president," he said.
Russia is under heavy Western sanctions and faces rising inflation.
In recent days, the ruble has been weakening sharply against the euro.
"Inflation is tough. What can we do?" said Peshcherkin, calling the economic difficulties "temporary".
During the Second World War "we had nothing to eat... and people survived... We still have something to eat," he said.
"Of course it is not at all good when there is a war, when people die," he said as an icy wind blew through the Russian capital.
"Everyone just has to be sensible and get around a negotiating table and talk," he said.
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