The tour starts Wednesday with an early London premiere, moving on to Paris and ending in New York.
Film stars are used to jetting around to launch their projects but not usually on the same day.
Blunt, the 31-year-old actress known from "The Devil Wears Prada," says it is typical of her co-star.
"When I heard about the premiere idea of three in one day I just said 'that's Tom's idea,'" she says.
"Edge of Tomorrow" is a combination of action, science-fiction, a war movie and a love story, directed by Doug Liman.
Cruise, 51, plays military PR man Major William Cage, a quick-talking coward who is forced to fight for humanity against an alien invasion. Blunt is the poster girl for the army, a highly-trained warrior and the only person who can help him when he starts living the same day over and over again, never reaching the tomorrow of the title.
Shot in the U.K. at Warner Brothers Leavesden Studios, director Doug Liman says three premieres in twenty-four hours is nothing compared to what they achieved on set.
"'Edge of Tomorrow' was an impossible movie to make - harder than anything I've ever done before. So, having a premiere day that starts off sounding impossible, really seemed like it would be a great way to finish," Liman says.
Cruise and Blunt joke about who will have more stamina staying awake on the day.
"Because I have a baby, I'm not sleeping as much anyway," Blunt says.
"You'll be like unconscious. But I'll be there," Cruise counters.
Cruise and Blunt were talking on the rooftop of a London hotel, overlooking the opening location of the movie, where the main character is flown in to Trafalgar Square.
It was the first time a helicopter has ever landed in between the famous fountains, next to Nelson's Column - something Cruise still gets excited talking about.
Despite shooting around the clock, the stars took time out when Blunt invited Cruise to a trendy London sex club to celebrate Matt Damon's birthday - a story she revealed on Jimmy Kimmel's U.S. chat show.
Cruise laughs off the fact that she has now told everyone.
"He thought it was going to be a more civilized night," Blunt says. "But it was fun."
'Edge of Tomorrow' is in U.K. cinemas on Friday and opens in the U.S. on June 6.