Israel said Hezbollah fighters had been embedded in tunnels under the ridge, which is one of the most hotly contested areas in the Israel-Hezbollah war, despite a ceasefire announced in June.
Iran warned the United States last month it would respond forcefully to an Israeli assault on its Hezbollah allies' positions within the ridge.
"The mission was completed and the tunnels were cleared of terrorists," Katz said on social media, after a statement by the military late on Thursday saying it had operational control of the ridge.
The military said some Hezbollah fighters had been "eliminated", while others had fled.
A Lebanese military officer said Israeli troops had seized the southern entrances to the complex and could monitor the northern entrances through surveillance drones.
The complex included various tunnels and storage areas, some of which were not connected to one another, and it was unclear how far the troops had penetrated, and whether any fighters were still there, or might have booby-trapped parts of the complex, the officer said.
Hezbollah did not immediately respond to a request for comment and Reuters was not able to verify the status of the ridge independently.
The Shi'ite Muslim group is believed to have built an underground command centre and weapons stores within the ridge and had said it would resist any Israeli assault.
Israel has expanded its military presence across swathes of southern Lebanon since launching a new campaign against Hezbollah in March.
On Friday, Israeli authorities transferred four people who had been detained in Israel to Lebanese authorities via the International Committee of the Red Cross (ICRC).
One Lebanese man had been released earlier on Thursday.
The release of the detainees is part of confidence-building measures agreed in US-brokered talks between Lebanon and Israel.
Israel said on Thursday none of the five had been found to have ties to militant groups.
A committee advocating for Lebanese citizens in Israeli custody says Israel has detained dozens of people inside the country over the last six months. Some have been released after days or weeks, others remain in custody.
The ICRC said on Friday it had not had access to anyone in Israeli detention since October 2023.
"Many families in Lebanon are still waiting for answers about the fate of loved ones who may be detained, deceased, or otherwise unaccounted for," it said.
Lebanon's President Joseph Aoun said on Friday those freed this week were among a "first batch", suggesting more could be coming.
Israel's Prime Minister Benjamin Netanyahu said Lebanon would transfer the remains of Jews killed in Lebanon in the 1980s to Israel, although there were no details of names or any timeframe.
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