The rescue by HMS Bulwark on Thursday was the first by a British ship since improved conditions allowed smugglers to begin operating again in March.
The migrants were transferred to an Italian coastguard vessel, Fiorillo, and the dinghy was sunk.
More than 1,800 migrants have died this year in the Mediterranean, a 20-fold increase on the same period in 2014.
A Ministry of Defence spokesperson said: "HMS Bulwark, working with the Italian coastguard, today investigated a large rubber boat with approximately 110 individuals on board.
"The individuals were rescued using Bulwark's landing craft before being transferred onto the Italian coastguard's vessel and taken to land. Everyone was transferred safely and HMS Bulwark remains on task."
The Bulwark sailed to the region earlier this month to join the rescue force and was declared operational on Monday.
Based at Naval Air Station Sigonella in Sicily, the ship will work with three Royal Navy Merlin helicopters to provide a "wide ranging search and rescue capability", the Navy said.
The Italian navy's search and rescue mission, Mare Nostrum, was abandoned last year after some EU members - including the UK - said they could not afford to fund it.
The UN says 60,000 people have tried to cross the Mediterranean in 2015. Many more are expected to attempt the crossing in the coming weeks as smugglers take advantage of calmer weather.
Italian and French Navy vessels rescued more than 5,800 migrants and recovered 10 bodies over the weekend from 17 different boats.
Italian authorities have estimated that as many as 200,000 people will attempt the journey before the end of the year, 30,000 more than in 2014.
A Royal Navy ship has rescued 110 migrants from a dinghy which was about to sink off the coast of Libya.