The offensive appeared to have surprised Russian troops and left them unprepared to defend their positions. Ukraine claims it has captured over 8,000 sq km (3,000 miles) of territory.
But Mr Putin said he has no plans to revise his goal of “liberating” the entire eastern Donbas region despite the successful counter-attack.
“We aren’t in a rush,” the Russian leader said, adding that Moscow has only deployed volunteer soldiers to fight in Ukraine.
Speaking at a summit in Uzbekistan, he said: “Let’s see how it develops and how it ends.”
He noted that Ukraine has tried to strike civilian infrastructure in Russia and “we so far have responded with restraint, but just yet.”
“If the situation develops this way, our response will be more serious,” Putin said.
“Just recently, the Russian armed forces have delivered a couple of impactful strikes,” he said in an apparent reference to Russian attacks earlier this week on power plants in northern Ukraine and a dam in the south. ”Let’s consider those as warning strikes.”
He alleged, without evidence, that Ukraine has attempted to launch attacks “near our nuclear facilities, nuclear power plants,” adding that “we will retaliate if they fail to understand that such methods are unacceptable.”
It comes after Mr Putin was publicly scrutinised over his invasion of Ukraine by India’s prime minister, who told him now “is not an era for war”.
Narendra Modi said he had “spoken to [Putin] on the phone” about the need to end the war, as the two met in Uzbekistan.
Mr Putin replied that he “understood” Mr Modi’s concerns and wished to end the war as soon as possible.
“I know your stand on the conflict in Ukraine and the concerns that you have repeatedly voiced,” the Russian leader told Modi.
“We will do all we can to end that as quickly as possible. Regrettably, the other side, the leadership of Ukraine, has rejected the negotiations process and stated that it wants to achieve its goals by military means, on the battlefield.”