He said that leading nations would now be allowed to assess each others' economic policies.
Mr Obama added that the leaders had agreed rules to ensure that executive pay would be linked to long term financial performance.
Many have criticised excessive bonuses as encouraging the kind of short term risk-taking that contributed to the financial crisis.
Despite Mr Obama's declaration, the G20 fell short of agreeing specific rules on the capital reserves that banks need to hold.
"We commit to developing by end-2010 internationally agreed rules to improve both the quantity and quality of bank capital and to discourage excessive [borrowing]," a statement from the G20 leaders said following the summit.
It added that the rules will be phased in once financial conditions improve and recovery is "assured".
The leaders also fell short of agreeing a cap on bonuses, agreeing instead that bonus payments should not be guaranteed for many years, should be deferred in part, and should not exceed a percentage of the bank's revenue.
"We designated the G20 to be the premier forum for our international economic co-operation," the statement said.
The lesson of Pittsburgh is that the old world still can't bring itself to follow through on its inclusive rhetoric.
It added that the global leaders would shift "at least 5%" of the quota of votes within the IMF from "over-represented countries to under-represented countries".
It described under-represented countries as "dynamic emerging markets and developing countries".
Emerging economies will also get a greater say at the World Bank.
The leaders also pledged to continue pumping money into their economies until "a durable recovery is secured".
But there will be no formal announcement that the G20 will replace the G8 until 2011, said the BBC's economics editor Stephanie Flanders.
"The leaders would have liked formally to announce the handover today in Pittsburgh, but the Canadians - who are chairing the G8 next year - kicked up such a fuss that they had to fudge it," she said.
There will now be a G20 meeting on the sidelines of Canada's G8 Summit next June, where most of the economic business of the day will be discussed.
But, formally at least, the economic side of the G8 will live on another year.
The IMF has 186 member-states. It lends money to countries that are facing problems, but in return economic changes have to be made by those countries.
Currently, China wields 3.7% of IMF votes compared with France's 4.9%, although the Chinese economy is now 50% larger than that of France.
The IMF has been criticised in the past as being a group of developed countries trying to lay down the law to struggling countries, which is why the decision to give growing nations more votes is important.
"If you talk to the Chinese or talk to anyone from emerging markets they say the IMF doesn't have legitimacy and... we don't trust the IMF to come and rescue us in a crisis," Simon Johnson, former chief economist at the IMF, told the BBC.
"They don't trust it because it's US and West Europe-dominated. That's not fair... and the IMF doesn't function properly as a result."
Nobel prize-winning economist Amartya Sen welcomed the change in voting rights, but said that, "on their own, they won't be able to achieve much... It's not just a question of voting rights, but also a question of broadening the dialogue".
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