A war monitor reported one person killed, with Syrian state media saying the strike had targeted a building in the capital.
Islamic Jihad fought alongside Hamas against Israel in Gaza before a fragile truce began in January.
Israeli Prime Minister Benjamin Netanyahu vowed that such strikes would be carried out across the region against "whoever attacks us".
"There will be no immunity for Islamic terrorism against Israel," Katz said in a statement. "We will not allow Syria to become a threat to the state of Israel."
The Israeli military said the "command centre was used to plan and direct terrorist activities by the Palestinian Islamic Jihad" against Israel.
A source in Islamic Jihad said a building belonging to the group had been hit by Israeli jets, adding there were "martyrs and wounded" in the strike.
Ismail Sindawi, Islamic Jihad's representative in Syria, told AFP the targeted building had been "closed for five years and nobody from the movement frequented it". Israel was just sending a message, Sindawi said.
Syria's official news agency SANA reported that three civilians were wounded, including one woman in a critical condition.
It said the strike targeted an office that had been "abandoned since the liberation of Damascus", when Islamist-led rebels toppled president Bashar al-Assad in December.
An AFP photographer saw the facade of the three-storey building completely destroyed and flames coming out from a balcony.
Netanyahu vowed to carry out more such strikes if needed.
"We attacked an Islamic Jihad headquarters in the heart of Damascus. We did this because we have a clear policy: Whoever attacks us or plans to attack us -- we strike them," Netanyahu said in a video statement.
"And this applies not only in Syria but everywhere, including Lebanon," where Israel and Iran-backed Hezbollah engaged in more than a year of hostilities until a ceasefire was reached in November.
- Hundreds of air strikes -
On Thursday evening, the Israeli military said it had carried out a strike in eastern Lebanon.
"A short while ago, the IDF (military) conducted a strike on infrastructure at a site used by Hezbollah for manufacturing and storing strategic weapons in the Beqaa area in Lebanon," the military said in a statement.
In November, Israel and Hezbollah agreed to a US-French mediated truce that has largely halted more than a year of fighting between the two sides, including two months of full-blown war in which Israel sent in ground troops.
While the ceasefire continues to hold, Israel has periodically carried out air strikes in Lebanon that it says are to prevent Hezbollah from rearming or returning to the area along its northern border.
Since Assad's overthrow, Israel has also carried out hundreds of air strikes in Syria and deployed troops to a UN-patrolled buffer zone on the strategic Golan Heights.
Netanyahu has previously said southern Syria must be completely demilitarised, warning that his government would not accept the presence of the forces of the new authorities near its territory.
The strike in Damascus came just before Syria's leader Ahmed al-Sharaa hailed the start of a "new history" for his country, signing into force a constitutional declaration regulating a five-year transitional period and laying out rights for women and freedom of expression.
Israeli Foreign Minister Gideon Saar has repeatedly warned that world leaders should be wary of the new leadership in Syria, warning that a "jihadist group" was now ruling the country.
Sharaa was the head of the Islamist group Hayat Tahrir al-Sham (HTS), which led the toppling of Assad and has its roots in the Syrian branch of Al-Qaeda. It has since sought to moderate its image, but is still listed as a terrorist organisation by the United States and other governments.
After years of diplomatic isolation under Assad, diplomats from the West and Syria's neighbours have reached out to Syria's new rulers.
Even before Assad's fall, during the Syrian civil war that broke out in 2011, Israel carried out hundreds of strikes in the country, mainly on government forces and Iranian-linked targets.
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