The stars are expected to hit the red carpets in London, Paris and New York May 28.
This is the first time three fan premieres will be held in three different countries in just one day, the studio said.
The stunt is in keeping with the film's plot, which sees Cruise's character reliving the events of one day over and over in an epic battle to save the world.
The first screening will take place in London at 9 a.m., while the showing in Paris is set for 4 p.m. The final premiere will take the stars back to the United States with a screening scheduled for "the edge of tomorrow" at 11:59 p.m.
"Cruise, Blunt and the film's director, Doug Liman, will interact with fans via social media as they jet from country to country. A contingent of select press will travel to each premiere, covering the events as they unfold," a news release said. "There will be live-stream components for each of the three events. Those fans on the ground can network via social media using #EOTLive."
Co-starring Bill Paxton, Brendan Gleeson, Noah Taylor, Kick Gurry, Dragomir Mrsic, Charlotte Riley, Jonas Armstrong, Franz Drameh, Masayoshi Haneda and Tony Way, the film is to open in U.S. theaters June 6.