In a nationally televised address, Brown said that Britain's message to those who threatened its security must be: "We will not yield, we will not be intimidated, and we will not allow anyone to undermine our British way of life."
The police arrested three men following Saturday's events and linked the attack with two possible car bombs discovered in central London a day earlier.
Two more people were also arrested on a motorway in northern England.
Wearing white plastic bodysuits and face masks, police in Glasgow raided houses near the airport on Sunday, about 10km west of the city, in the town of Houston in the Renfrewshire area.
Neighbours said two Asian men had moved into one of the searched houses about a month ago but had kept very much to themselves.
"I don't remember seeing them at all," said Mae Gordon. "They were the only people around here you would never see."
Later on Sunday, police said they had carried out a controlled detonation of a suspicious vehicle left in the car park of a hospital near Glasgow where one of the two airport assailants was being treated for severe burns.
They said they believed the car was linked to the attack on the airport, but said it was not thought to contain explosives.
Britain has seen an increase in attacks since it joined US forces in invading Iraq in 2003.
Analysts say the latest attacks may be designed to exert pressure on Britain to withdraw its troops from Iraq and Afghanistan.
Brown, who took over from Tony Blair on Wednesday, called a third meeting of Britain's Cobra committee, a national crisis response team, to discuss measures to handle his first major challenge in office.
Speaking on BBC television on Sunday, Brown warned the "fight against terrorism" would be drawn out.
"Irrespective of Iraq, irrespective of Afghanistan, irrespective of what is happening in different parts of the world, we have an international organisation trying to inflict the maximum damage on civilian life in pursuit of a terrorist cause that is totally unacceptable to most people," he said.
|Five people have been arrested over the |
incidents in London and Glasgow
In response to the events in the UK, the White House announced that security was being stepped up at US airports, although a spokesman said there was "no indication of any specific or credible threat to the United States".
The authorities did briefly close the American Airlines terminal at New York's John F Kennedy International Airport on Sunday after a suspicious package was found.
Port authority officials said the package was being investigated by the bomb squad and other experts but the terminal was re-opened and the "all clear" given shortly after it was evacuated, media reports said.
No information was available on what was inside the package. The airline said flights were not delayed.
The developments came a week before the second anniversary of the July 7, 2005 suicide bombings in London which killed 52 people during the morning rush-hour.