He said Suri was a Syrian national and a "legacy" Al-Qaeda member who fought in Afghanistan in the late 1980s and 1990s.
He "worked with Osama bin Laden and other founding Al-Qaeda members to train terrorists and conduct attacks globally," Cook said, adding that Sunday's strike killed several enemy fighters.
According to the Syrian Observatory for Human Rights, Suri, his son and at least 20 jihadists of Al-Nusra and Jund al-Aqsa and other fighters from Uzbekistan were killed in strikes on positions in Idlib province.
Seven were high-ranking jihadists, the Britain-based Observatory said.
Aymenn Jawad al-Tamimi, a research fellow at the Middle East Forum, a US think-tank, said Suri was a top jihadist official.
Suri "was a very senior member of Al-Nusra, but organisations like Al-Nusra aren't debilitated because they lose a single senior leader", he said.
"Their organisational structures are well prepared for targeted assassinations, which are usual business for them."
Suri, whose real name was Radwan Nammous, fought against Soviet forces in Afghanistan where he met Bin Laden and the founding father of global jihad, Abdullah Azzam, before returning to Syria in 2011.
- Warning to Al-Nusra? -
A temporary ceasefire between government forces and rebels has largely held since February 27, but it does not cover Al-Nusra and IS.
The break has allowed Russia and the US-led coalition that has been bombing IS in Syria to concentrate on their fight against the jihadists.
Al-Nusra has generally kept a low profile since the truce brokered by the United States and Russia came into force.
But on Friday, the Al-Qaeda affiliate and allied rebels pushed regime loyalists out of Al-Eis, a strategic town in the northern province of Aleppo, killing 12 members of the Lebanese Shiite Hezbollah movement.
"It was Al-Nusra's biggest operation since the ceasefire began," Observatory chief Rami Abdel Rahman said.
IS has also lost a string of high-ranking members in recent weeks, mainly to strikes by the US-led coalition that launched an aerial campaign against the jihadists in Iraq and Syria in 2014.
Last Wednesday, a drone strike near IS's de facto capital Raqa killed Tunisian commander Abu al-Haija.
Fifteen IS commanders accused of revealing his position have since been executed by the jihadists, and the fate of another 20 men accused of collaborating with the US-led coalition remains unknown.
"This is the highest number of executions of security officials by IS," said Abdel Rahman, whose group relies on sources in Syria for its reports.
- Recapture of Al-Qaryatain -
On Monday, IS's press officer in the eastern province of Deir Ezzor was killed in an air strike while covering fighting between the jihadists and regime troops, the Observatory said.
"It was unclear whether the air strike that killed Mohammad al-Lafi was Russian or Syrian," the group said, adding that the IS official used the nom de guerre Abu Abdallah Azzam.
On Sunday, the army seized the town of Al-Qaryatain, one of the last IS strongholds in central Syria, a week after the Russian-backed army scored a major victory in the ancient city of Palmyra, also located in the vast province of Homs.
The recapture of Al-Qaryatain allows the army to secure its grip over Palmyra, where jihadists destroyed ancient temples during their 10-month rule and executed 280 people.
It has also left IS with just one bastion in Homs province, Sukhna, where the focus of the fighting has now shifted.
In spite of the truce, hundreds of thousands of civilians living under siege across Syria remain deprived of essential medical and food assistance, according to Human Rights Watch.
"While aid delivery has improved in the last month, it's still not nearly enough and too many Syrians are still not receiving the aid they need," said Nadim Houry, HRW's deputy Middle East director.
Russia's renowned Hermitage Museum, which has an important collection of sculptures from Palmyra, said Monday it was willing to help restore the ancient Syrian city.