"We must root out terrorism and extremism ... [but] we will not tolerate the violation of our sovereignty and territorial integrity by any power in the name of combating terrorism," he said.
Curb on powers
He also called for a parliamentary committee to be formed to cut back his presidential powers, including those allowing him to dissolve the assembly and dismiss the government.
"Never before in the history of this country has a president given away his powers," Zardari said.
He called for the committee to "revisit" the constitutional amendments made by his predecessor, Pervez Musharraf, who stepped down in August after the coalition government threatened to impeach him.
Zardari said in future, the presidency and the government should follow parliament.
"As head of the state I wish to make it very clear that the president and the government must always seek guidance from the parliament in carrying out our duties," he said.
Kamal Hyder, Al Jazeera's correspondent in Islamabad, said: "The nation has a lot of expectations. They want to hear from their new president [reasons] why he has been silent about the American incursions and whether he is on board as far as the mood in the country is concerned.
"Of course, everybody will be anticipating to hear all the right things, whether the president will be able to deliver on those or not is something that we will have to wait for," he said.
Zardari, the widower of the assassinated Benazir Bhutto, a former prime minister and leader of the Pakistan Peoples Party, was sworn into office on September 9.
The man he replaced, Musharraf, addressed the parliament only once during his eight-year rule and on that occasion opposition politicians chanted hostile slogans.
When the government pushed Musharraf to resign, one of their complaints was his failure to address parliament on an annual basis, in line with his constitutional commitments.