Hezbollah operative Ali Musa Daqduq is the network's commander and has been operating covertly and apparently without the knowledge of Syrian President Bashar Assad's government, the Israeli military said.
"These operatives have been attempting to establish an additional terrorist front against Israel and destabilize the region over the last year," said Brig. Gen. Amit Fisher, the commander of one of Israel's northern divisions.
He warned that Israel will "not allow any attempt by Hezbollah to entrench itself near the border and we will act with all our might."
There was no immediate comment from Hezbollah.
Israel considers Iran to be its greatest enemy and has repeatedly warned that Tehran is trying to entrench itself militarily in Syria as that country's civil war winds down. Iranian and Hezbollah fighters have sided with Assad's forces in the war and are, along with Russia, the Syrian president's biggest allies.
While Israel has largely stayed out of the fighting in Syria, it has carried out hundreds of airstrikes on suspected Iranian arms shipments to Hezbollah in Lebanon. Israel has recently been more public about its military actions and has openly warned of the consequences of Iran establishing a permanent foothold in post-war Syria.
The Israeli military said the uncovered network was in its early stages and not yet operational.
Israel recently exposed a Hezbollah tunnel network from Lebanon it says was aimed at carrying out attacks in Israel.