The US and Afghanistan say al-Qaeda and Taliban fighters are hiding in sanctuaries in northwest Pakistani border areas, from where they orchestrate fighting in Afghanistan and Pakistan and plot violence towards the West.
Bush said the fighters were stepping up their attacks from the sanctuaries and that he would be sending more US troops to Afghanistan to combat the threat.
Speaking at the swearing-in ceremony in Islamabad, Zardari said: "We shall stand with our neighbours ... and look the problems in the eye and tell the world that we are bigger than the problems."
Hamid Karzai, the president of Afghanistan, and Zardari's three children attended the inauguration in the country's capital.
Zardari later told a news conference held with Karzai that the government was following a comprehensive plan on tackling fighters and both presidents stressed their would work together to fight the problem.
The Pakistan president said his election had provided democratic impetus to the government's "fight against terrorism".
"The government of Pakistan already has a comprehensive plan. Yesterday's war may not have had the people behind it, but today's war has the people behind it, it has the Pakistani president behind it, who himself is a victim of terrorism," he said.
Karzai reiterated his call to go after the fighters in their sanctuaries, whether in Pakistan or Afghanistan, and for international forces to avoid civilian casualties after a spate of such incidents in which scores of Afghans were killed.
He said: "The war against terrorism will only be won if we have the people with us. In order to have the people with us we must avoid civilian casualties."
Zardari's party heads a fragile coalition government which, although still in office, recently lost the backing of a key coalition party.
He takes office as many Pakistanis are furious with the US after a bloody incursion by its ground troops into a remote village on the Afghan border last week and a string of missile strikes by CIA-operated drone aircraft.
Zardari said the government had protested to the US and that "casualties of war are taking place. We cannot deny that innocents are dying".
Investors are hoping Zardari's election win will end political uncertainty that has dragged stocks and the rupee sharply lower.
The Karachi Stock Exchange benchmark share index ended marginally lower, with analysts saying investors wanted to see concrete action being taken to bolster the economy.
Zardari said the government would not seek an International Monetary Fund programme, but would take advice from the organisation.
The president said he hoped his first foreign trip would be to China and he would attend a UN general assembly session in New York.
Zardari's decision in August to begin impeachment proceedings against Musharraf, led to the former president's resignation.
The president said parliament would decide on whether Musharraf should get an indemnity from any prosecution.