British and Canadian experts say drugs could attack an enzyme the parasite causing the illness needs to survive.
They say the orally-administered drug could be ready for human clinical trials in about 18 months.
The disease, spread by the bite of a tsetse fly, is caused by a parasite attacking the central nervous system.
It has similar symptoms to malaria, making it difficult to diagnose. Left untreated, it moves to the spinal column and brain, resulting in mental confusion and eventual death.
Fatal side effects
The “breakthrough” came at the University of Dundee in Scotland, where scientists were funded to research diseases neglected by major drugs companies.
Professor Paul Wyatt, director of the programme, said: “This is one of the most significant findings made in recent years in terms of drug discovery and development for neglected diseases.”
He said the research, published in the journal Nature, represented “significant strides” in the development of a full blown drug against the disease.
The World Health Organization estimates there are between 50,000 and 70,000 cases of the disease a year, with a further 60 million people at risk of infection.
The research in Dundee was backed by partners at the University of York in England and the Structural Genomics Consortium in Toronto, Canada.
The two drugs currently available to treat sleeping sickness both have associated problems.
One is arsenic-based with side effects that kill one in 20 patients and the other – eflornithine – is costly, only partially effective and requires prolonged hospital treatment, the scientists said.