And Mr Brown hailed the US President as "a true friend of Britain" who had enhanced the transatlantic partnership between the two nations.
Despite reports of unhappiness at the White House over British plans for troop withdrawals from Iraq, the two leaders insisted they shared the goal of bringing soldiers home as they were able to hand over security responsibility to local forces.
Mr Bush welcomed Mr Brown's announcement of additional British troops for Afghanistan, as well as his call for tougher European Union sanctions on Iran.
And he pointed out that the UK's 4,000-strong deployment in Iraq remained higher than anticipated when Mr Brown last autumn announced an intention to cut it to 2,500.
The mutual praise at a press conference in the Foreign Office came in marked contrast to the pair's first meeting after Mr Brown's elevation to the premiership a year ago, when the PM was perceived to have attempted to distance himself from Mr Bush.
Today's trip is expected to be Mr Bush's last official visit to the UK before he hands over to his successor in the New Year. But he teased reporters by saying: "Some are speculating this is my last trip. Let them speculate - who knows?"
Mr Bush said that Mr Brown had been "tested" by the Heathrow plane plot soon after entering Number 10 last summer and had responded to the challenge.
"The first thing about Gordon Brown, he is tough on terrorism, and I appreciate it," he told journalists.
"So should the British people... You have been strong on Afghanistan and Iraq and I appreciate that. But more importantly, the people of Afghanistan and Iraq appreciate it."
Mr Brown thanked the President for the "steadfastness and resolution that he had shown in rooting out terrorism in all parts of the world".
Mr Bush was a "true friend of Britain", and today's visit offered a chance "to celebrate the historic partnership of shared purpose that unites the UK and the USA", he added.
Mr Brown said: "Let me thank Mr Bush for being a true friend of Britain and for the importance he attaches to enhancing our transatlantic partnership for the work we do in Afghanistan and Iraq and every part of the world.
"Let me thank him for the steadfastness and resolution he has shown in rooting out terrorism in all parts of the world, in working for a Middle East peace settlement, in bringing hope to Africa, in working for a free trade world where - in spite of today's difficulties with oil and food prices - there is and should be a wider and deeper prosperity in future for all."
He and Mr Bush had agreed that there was a "special partnership" between the governments and people of Britain and the US "driven forward not simply by mutual interests but shared values" of liberty, democracy and opportunity for all, Mr Brown said. Tweet
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what about being tough on crimimals in the UK and immigration and the cost of flipping living. Get real.
These would be the same troop withdrawls that Gordon Brown cynically announced to try and deflect attention from his conference speech.
I wonder what he will try this year? Maybe he should try and deflect away from his own speech because nobody is listening anymore. We are all counting down to 2010
It must have been an important meeting, Richard Leese only got an hour at Number 10.
It's great to have such a powerful ally.
Let's move on to even better things and turn Europe into a super-power.
As for "tough on terrorism", he should start by deporting the mad mullahs who stir up all the trouble.
puppet on a string springs to mind, one min were gonna start pulling troops out next min were sending more in!!
talks about oil prices were mentioned what happens oil per barrel goes up yet again.
lets get this farciacal government out, were paying more tax than weve ever paid before.
We have america to thank for getting us involved in americas new (vietnam) this war will rage on for years.Russia was fighting a war with these people for years will a million troups and didnt win.We have brave lads fighting this war for what at the end? Its ok for these politicians to look smug but when we send our lads to war with very little equipment.very anoyed at our politicians.