In fact, it was truly a series of events that got Brando cast in the role.
The film's director, Francis Ford Coppola, was dead set on casting either Brando or Laurence Olivier as Vito. When he was told no to casting Brando by Paramount’s then-president, Stanley Jaffe, Coppola faked a stress-induced epilepsy in Jaffe's office. After that stunt, Jaffe reconsidered and told him he would consider Brando for the role on three conditions — with one of them being that he would need to screen-test for it.
Meanwhile, Mario Puzo, who wrote The Godfather novel, read a story that comedian–sitcom star–producer Danny Thomas was interested in buying a majority share in Paramount with the sole purpose of casting himself as Vito. That caused an alarmed Puzo to write a letter to Brando — whom he had envisioned in the role — and plead with him to consider the role of Vito.
Brando's personal secretary, Alice Marchak, knew that he needed to work because he was deeply in debt, so she bought a subscription to the Hollywood Reporter to keep tabs on potential roles for him. However, he caught her reading it one day and got mad that she would bring "movie magazines into [his] house." So she left his house and let him cool off for a few days, during which time she read that Paramount was looking to cast the role of Vito in The Godfather. When she came back to Brando's house, she found Puzo's letter in the mail.
Puzo and Brando would eventually talk over the phone, and by then Marchak had read the book and knew the role was right for him. She also slowly played to Brando's ego by commenting to him about all his fellow actors who were being considered for the role — causing him to want the part even more.
Eventually he agreed to it, but Coppola knew he would need to screen-test him in order to cast him (something that would be an insult to an actor of Brando's stature). So he devised a plan in which he didn't tell Brando it was a screen test, but instead told him that he wanted to come over to his house to try some things for the camera, experiment a bit, and try some improvisation. And that worked.
When Coppola finally did the not-a-screen-test-at-all at Brando's house, he saw the 47-year-old actor transform into Vito in front of his eyes as he smeared black shoe polish on his blonde hair to give it a darker and greased-back look, gave himself a shoe polish mustache, and put wads of tissue in his cheeks to give himself an appearance of a bulldog. The rest, of course, is movie history.