The drones had crossed into Jordanian territory and their hauls of crystal methamphetamine were seized, the army said on Tuesday.
It warned it would act forcefully to prevent any attempt to destabilise the country.
Jordanian officials say the increasing use of drones carrying drugs, weapons and explosives is adding a new dimension to a cross-border billion-dollar drug war the US ally has blamed on Iranian-backed militias that hold sway in southern Syria.
Syria is accused by Arab governments and the West of producing the highly addictive and lucrative amphetamine Captagon and other drugs, and organising its smuggling into the Gulf, with Jordan a main transit route.
Syrian President Bashar Al Assad's government denies allegations of his country's involvement in drug production and smuggling, as well as any complicity by Iranian-backed militias protected by units within the Syrian army and security forces.
Iran says the allegations are part of a western plot against the country.
Jordanian officials say talks with senior Syrian officials to curb Iranian-run smuggling networks have reached a dead end due to the inability of Damascus to impose order over its southern region.
Jordan's King Abdullah said last week that Iran and elements within the Syrian government were benefitting from the drug trade. He added that he was not sure if Mr Al Assad was fully in charge of the country in view of the “major problem” of drugs and weapons being smuggled.
“We are fighting every single day on our border to stop massive amounts of drugs coming into our country,” King Abdullah said at the Middle East Global Summit conference in New York.
“And this is a major issue that all the parties, including some people inside the (Syrian) regime, and the Iranians and their proxies, are all taking advantage of.”
Foreign Minister Ayman Safadi also said the amount of drugs being smuggled from Syria had increased after the kingdom launched talks with Damascus on curbing it since the sanctions-hit country returned to the Arab fold last May.
During a visit last month by Gen Mark Milley, the chairman of the US Joint Chiefs of Staff, Jordan requested more support for its efforts to curb drug trafficking, Amman officials say.
Mr Milley confirmed that Washington was working closely with its ally to provide equipment, training and advice to deal with the growing drug trafficking threat.