Self-proclaimed Pakistani Taliban leader Baitullah Mehsud and other Taliban officials in recent weeks have vowed to send fighters into Afghanistan to battle Afghan and foreign forces.
"This means that Afghanistan has the right of self defense," Karzai said, when asked to comment on the threats.
"When they cross the territory from Pakistan to come and kill Afghans and kill coalition troops, it exactly gives us the right to go back and do the same. Therefore, Baitullah Mehsud should know that we will go after him now and hit him in his house."
Pakistani officials and Taliban spokesmen were not immediately available for comments.
Pakistan, wary of a backlash from Pashtun tribes living on its long border with Afghanistan, has ruled out allowing foreign troops to operate on its territory, though U.S. forces have carried out several airstrikes inside Pakistan targeting al Qaeda militants and their allies.
Eleven Pakistani paramilitary soldiers were killed in an air raid by U.S.-led coalition forces against the militants on the border this week.
Both Afghanistan and Pakistan, who historically have uneasy ties and border disputes, are allies of the United States in its war against al Qaeda and the Taliban.
Nearly 13,000 people, including hundreds of foreign troops, have been killed in Afghanistan since 2006 when the ousted Afghan Taliban made a come back.
The Afghan government says the Taliban and al Qaeda members who fled into Pakistan after the fall of the militants' government in Afghanistan in 2001, have support and training bases there.
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Pakistan concedes some cross border infiltration of militants but wants Afghan government and foreign forces to tighten border controls on their side.
Pakistan, reeling from a wave of suicide attacks since mid-2007, has opened peace talks with militants, including Mehsud. Afghan and NATO officials say cross-border attacks have risen since then.
Pakistan supported the largely ethnic Pashtun Taliban militia when it seized control of Afghanistan in the mid-1990s and officially dropped its support after the September 11 attacks on the United States.
Karzai, an ethnic Pashtun who is facing an election next year, said Mehsud and other Pakistani Taliban were tools of Pakistan's intelligence services.
"(They) have been trained against the Pashtuns of Pakistan and against the people of Afghanistan and their jobs are to burn Pashtun schools in Pakistan, not to allow their girls to get educated and kill the Pashtun tribal chiefs," he said.
"This is the duty of Afghanistan to save the Pashtuns in Pakistan from this tyranny and terror ... this is the duty of Afghanistan to defend itself and defend their brothers, sisters and sons on the other side."
There is a long-running dispute between Pakistan and Afghanistan over Pashtun territory straddling both sides of the border. Afghanistan has never recognized the 2,640 km (1,610) frontier, known as the Durand line after the British colonialist who drew it. Afghans say the border robbed Afghanistan of land and unfairly divides Pashtuns.
Editing by Zeeshan Haider and Sanjeev Miglani; Editing by Janet Lawrence
© 2008 Reuters