Russian officials described Poroshenko's remarks as positive, and on his return to Moscow, Putin appeared on television to order his Federal Security Service to stop illegal border crossings. Ukraine and its Western allies have repeatedly called on Russia to stop armed groups moving into eastern Ukraine, where militias have seized key cities.
A billionaire universally known as the chocolate king, Poroshenko made an impassioned plea for the unity of his country.
Addressing the people of the eastern half of the country in Russian, he offered a peace plan to overcome divisions.
"I don't want war; I don't want revenge. I want peace and I want peace to happen," he said after taking the oath of office. "Please, lay down the guns and I guarantee immunity to all those who don't have bloodshed on their hands."
In a significant gesture, Russia sent its ambassador to his inauguration in Kiev.
Face-to-face, Poroshenko and Putin appeared to find some common ground on the dispute over territory, the rights of Russian-speaking Ukrainians and a stand-off over Russian supplies of gas to Kiev.
"The approach seems to me altogether right: he has pleased me," declared Putin. "I can't tell exactly how this will be implemented, but in general, I liked the attitude."
Poroshenko said the security of Ukrainians could not be guaranteed until the breach caused by Yanukovych's departure was repaired. "A Russian representative will travel to Ukraine, and we will discuss with him the first steps towards a plan [to resolve] the situation. We have a good chance of implementing it."
However, Russian-backed rebels in control of eastern cities dismissed his overtures. Valery Bolotov, governor of the self-proclaimed "Luhansk People's Republic" rejected Ukrainian rule.
"As for our republic, we have no diplomatic relations with Ukraine," he said. "Today, Ukraine got a new President, and now the blood of our people and of Ukrainians will lie on his conscience."
Biden son gets job in Ukraine
US Vice-President Joe Biden's visit yesterday to support Ukraine's fragile democracy came soon after his youngest son was hired by a private Ukrainian company that promotes energy independence from Moscow.
Yet that company leases natural gas fields in the breakaway Russian-backed state of Crimea and is owned by a former government minister with ties to Ukraine's ousted pro-Russian President.
The hiring of Hunter Biden, 44, by Burisma Holdings Limited in April was approved by the company's owner, a former senior minister and political ally of exiled Ukrainian president Viktor Yanukovych, who fled to Russia in February after protests erupted over his efforts to establish closer economic ties with Moscow.
Hunter Biden's employment means he will be working as a director and top lawyer for a Ukrainian energy company during the period when his father and others in the Obama Administration bid to influence the policies of Ukraine's new Government, especially on energy issues.
There's no indication that Hunter Biden, his father or Burisma are crossing legal or ethical lines, but ethics experts are divided over the implications of Hunter Biden's job.