A major search operation was launched in both England and France after some of the group phoned their families from the vessel and they alerted the authorities.
The UK Coastguard received a call for assistance just off the coast of Dymchurch in Kent, at 11.40pm on Saturday.
The rhib (rigid-hulled inflatable boat) – with 20 people on board – was found at 2am. The migrants were taken ashore to Dover and the case was handed over to Border Force officials.
A Home Office spokesman said: “A total of 20 people were picked up in a search-and-rescue operation. 18 were Albanian, and two were British.
“They were taken to Dover and are currently being interviewed by Border Force officers.”
Councillor Mary Lawes, Ukip group leader on Shepway District Council, said she was concerned for the security of the region as well as the safety of migrants seeking to cross the Channel in unsafe boats.
She said: “We are not doing enough to control our coastline, the Government has to address border controls, something has to be done to protect these people from harm and our borders.”
David Monk, Conservative leader of the local authority, said he believed high levels of surveillance in the English Channel would mean most boats crossing the channel would be identified.
He added: “I am pretty sure our security is good. I cannot recall a previous incident but this should act as a warning to the authorities to be even more vigilant.”
The incident comes after 17 suspected Albanian migrants and a British man wanted on suspicion of murder in Spain were detained after a catamaran arrived at Chichester Marina in West Sussex on Tuesday.
The 55-year-old man, who was the subject of a European Arrest Warrant, was also detained on suspicion of facilitating illegal immigration while the 17 Albanian men were held on suspicion of entering the UK illegally.
The Albanians have been detained pending Home Office consideration of their cases.
The National Crime Agency (NCA) recently revealed migrants trying to reach the UK are paying smuggling gangs up to £13,500 for their journey.
A Coastguard statement said: “The HM Coastguard search-and-rescue helicopter from Lydd, RNLI lifeboats from Dungeness and Littlestone, and coastguard rescue teams from Dungeness and Folkestone were sent.”