Mr Obama warned that countries should not "project inwards" by encouraging protectionism during the economic crisis.
Mr Brown, who will host a summit of the Group of 20 (G20) developed and emerging economies in London on 2 April, said: "I have come here to renew our special relationship for new times. It is a partnership of purpose born out of shared values.
"It is a partnership of purpose founded on determination to rise to every challenge and it is a partnership of purpose driven forward now by the need for all of us to work together to deal with the world's economic problems."
Mr Obama also urged coordinated action, saying: "All of these steps, I think, are going to slowly build confidence but it is not going to happen yet.
"We together have dug a very deep hole for ourselves. There were a lot of bad decisions that were made. We are cleaning up that mess.
"There are going to be fits and starts in getting the mess cleaned up but it is going to get cleaned up and we will emerge more prosperous, more unified and more protected from systemic risks."
BBC political editor Nick Robinson said the press conference had a "very different feel" to those more informal ones involving predecessors Tony Blair and George W Bush.
Michelle Obama and Sarah Brown also held a meeting
Downing Street dismissed suggestions that Mr Brown's visit had been downgraded by Mr Obama after it emerged there would not be a formal joint press conference.
The prime minister's wife Sarah also met First Lady Michelle Obama away from the media cameras.
Mr Brown's address to Congress is expected to compare the battle against the global recession to the fight against European fascism in the 1940s.
Former assistant secretary of state James Rubin said one of the current fears was protectionism.
He said: "Everyone who's an international economist is worried about this sort of thing. But everyone who's a politician knows that people are going to have to take care of their own in a time of crisis.
"That doesn't mean there isn't going to be plenty of trade and plenty of international economic cooperation. But when you have a crisis like this people have a tendency to look first to their own constituents."
The EU and Canada have warned that a "buy American" clause in the US economic recovery package could promote protectionism.
It seeks to ensure that only US iron, steel and manufactured goods are used in construction work funded by the bill - but has included a pledge to respect international trade obligations.
Former UK foreign secretary Sir Malcolm Rifkind said: "The single biggest issue is the issue of protectionism, maintaining the free trade system which has done so much to increase prosperity over the last 30 to 50 years and the President knows that. The President is unlikely to challenge that.
"But within Congress you have a large number of congressmen who don't have the responsibilities of government, who are under pressure from constituents and they are the people who have to be influenced, if they can be influenced, by Gordon Brown and his address."
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