The agency has 275,000 outstanding cases – equivalent to the population of Newcastle – a report by the Commons home affairs committee said.
Keith Vaz, the committee's chairman, said that the backlog would take years to clear and that the agency appeared to have "acquired its own Bermuda Triangle".
"It's easy to get in, but near impossible to keep track of anyone, let alone get them out," the Labour MP said.
The latest report on the agency's work said the committee did not believe cutting the 260,000 student visas issued each year by a quarter would benefit the UK. Students, a market worth £7.9billion, should be excluded from net migration figures, it said.
University chancellors have urged the Prime Minister to class international students as temporary migrants.