Libyan rescuers took three hours to reach the sinking rubber dinghy, officer Rami Ghommeidh said.
A survivor said they had been close to an Italian vessel which had refused to save them.
"When we got closer to the Italian rescuers, they wouldn't rescue us so we waited there" for the Libyan navy, he said.
Traffickers kill 12 migrants trying to flee Libya camp: UN
Panic ensued as the passengers sought to clamber on board the Libyan ship as their own boat was quickly filling with water.
"While trying to climb up the side of the rescue boat, three women and two little boys fell into the water and drowned," he said.
Ghommeidh did not identify the victims, confirming five people died and the 117 survivors included women and children.
90 migrants, mostly Pakistanis, believed dead in shipwreck off Libya
They were taken to the capital Tripoli after being rescued "around eight nautical miles off the coast of Abu Kammash", an important petrochemical complex and oil terminal close to the Tunisian border, the officer said.
Libya is a key departure point for thousands of migrants hoping to reach Europe, although hundreds drown each year attempting the sea route.
32 Pakistanis boarded capsized Libyan boat, 11 bodies recovered so far, clarifies FO
So far this year around 40,000 people have survived the crossing, while more than 800 are recorded as dead or missing by the United Nations refugee agency.
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