Unlike more traditional forms of cancer treatment that directly target cancer cells - often with severe side-effects - Allison and Honjo figured out how to help the patient's own immune system tackle cancer more quickly.
In 1992, Honjo discovered a protein on immune cells called PD-1.
"Allison and Honjo showed how different strategies for inhibiting the brakes on the immune system can be used in the treatment of cancer", it said.
It was a rare award for a key cancer breakthrough, in this case the discovery that the immune system can be tweaked to unleash tumor-attacking T cells. "I didn't set out to study cancer, but to understand the biology of T cells, these incredible cells [that] travel our bodies and work to protect us".
Also on Monday, a French photographer at the center of the scandal that led to the unprecedented postponement of the 2018 Nobel Prize in Literature was sentenced to two years in prison for rape. In 1996, Allison's team showed that antibodies against CTLA-4 not only got rid of cancer, but prevented new tumors from forming in mice.
The list of other possible awardees included a number of American researchers including Arlene Sharpe and Gordon Freeman at Harvard Medical School and Dana-Farber Cancer Institute; Jedd Wolchok at Memorial Sloan Kettering Cancer Center; and Carl June at the University of Pennsylvania, who pioneered another approach to immunotherapy.
Allison studied a protein that acts as a brake on the immune system and the potential of releasing that brake. Allison developed this idea into a new type of cancer treatment.
The therapy "has now revolutionized cancer treatment and has fundamentally changed the way we view how cancer can be managed", the statement added.
Allison, who is the chair of Immunology and executive director of the Immunotherapy Platform, is the first MD Anderson scientist to receive the world's most coveted award for discoveries in the fields of life sciences and medicine. "And I thought, 'If we could do that in people, this is going to be fantastic'".
"I've been doing this sort of stuff for years, and I'd never seen anything like that", Allison said.
In 2016, after being treated with a drug inspired by Prof Honjo's research, he announced that he no longer needed treatment.
The discovery made by the two laureates "constitutes a landmark in our fight against cancer", the committee tweeted shortly after the announcement.
Allison started his career at MD Anderson in 1977, arriving as one of the first employees of a new basic science research center located in Smithville, Texas.
No Nobel Literature Prize is being given this year because the Swedish Academy, the body that choses the literature victor, has been in turmoil after sex abuse and financial scandal allegations.
The prizes for physics, chemistry, and peace will also be announced this week.