The 48-year-old tycoon, who won the 25 May election, offered political concessions to people in the east and said he did not want war or revenge.
But he also said he had told Russia's president that Crimea, which Moscow has annexed, would "always be Ukrainian".
Russia's ambassador said the speech was a "promising declaration of intent".
Mikhail Zurabov, who attended the inauguration, said Ukraine should end the military operation in the east, provided that militias called a ceasefire and allowed humanitarian access.
Kiev says Moscow is backing armed militants in the eastern Donbass area, an accusation that Russia denies.
Mr Poroshenko was inaugurated in the presence of dozens of foreign dignitaries - including US Vice-President Joe Biden - in parliament in the capital Kiev.
Mr Poroshenko, the owner of the Roshen chocolates group, took the oath of office and laid out a programme for ending the crisis that included an offer of early regional elections in the east and a decentralisation of power to the regional administrations.
He said: "I don't want war. I don't want revenge, despite the huge sacrifice of the Ukrainian people."
Mr Poroshenko called on separatists in the east to lay down their arms, saying he would guarantee indemnity from criminal charges to all those who did not have blood on their hands.
But he added: "Talking to gangsters and killers is not our path."
The BBC's David Stern in Kiev says part of the speech was in Russian - directed at those in strife-torn Donbass - and offered concessions on Russian language and a corridor for Russian fighters to return home.
Petro Poroshenko's inauguration speech was forceful and seemed to hit all the right notes for his supporters.
Local commentators and bloggers' reaction was overwhelmingly positive.
Pro-revolution Ukrainians were undoubtedly heartened by his strong position on keeping the country unified, and advocacy for EU membership.
However, a number of groups have yet to be heard from: the pro-Russian separatists, officials in Moscow, and last but not least, the general population in the east.
The Kremlin and the insurgents will give their reaction fairly soon: Moscow and Kiev officials are expected to meet on Sunday to discuss the crisis in Ukraine's east.
But Ukrainians in the east and south may take a while longer to let their opinion be known.
Like their countrymen in the centre and west, they may see Mr Poroshenko as a man who will defend their interests.
But they may be angered by his calls to join the EU and insistence that Ukrainian remain the sole state language - and feel once again that their voices are not being listened to.
But our correspondent says there were also standing ovations for the president's comments on Crimea and on ties with the European Union - which he reiterated he wanted Ukraine to join.
Mr Poroshenko said of Crimea - annexed in March by Russia - "Crimea is, was and always will be Ukrainian soil."
Referring to a brief meeting with Russian President Vladimir Putin in France on Friday, he said:
"I put that clearly to the Russian leader in Normandy."
Mr Poroshenko also said there would be no discussion concerning the territorial integrity of Ukraine.
"I will make unity happen," he said.
Mr Poroshenko condemned the rule of former President Viktor Yanukovych, seen by many as pro-Russian, who fled in February after a popular uprising in Kiev.
He accused Mr Yanukovych of financing terrorism in the east, saying he was "fully responsible for the situation there today".
Clashes continued in some eastern areas as the inauguration loomed.
The BBC's Steve Rosenberg, in Donetsk, said there were reports on Saturday of fresh fighting near the rebel stronghold of Slaviansk and shooting further south in Mariupol.
On Friday, pro-Russian rebels reportedly shot down a government aircraft near Slaviansk.
After their meeting on Friday, Mr Putin said he liked Mr Poroshenko's approach but would wait to see what he could deliver.
"I can only welcome Mr Poroshenko's position that the bloodshed in eastern Ukraine must be stopped immediately," he said, adding that this had to mean an end to the Ukrainian government's "punitive operation".
"If it continues like that then conditions will be created for developing our relations in other areas as well."
Mr Putin and US President Barack Obama also held an "informal meeting" lasting about 10-15 minutes, according to the White House.
Targeted sanctions were introduced by the EU and US after Russia annexed Crimea, following a controversial referendum on joining Russia.
Since then, a bloody insurgency has gripped Ukraine's eastern provinces of Donetsk and Luhansk, on the Russian border.
Source: BBC News Europe