Landau, 89, died on Saturday of unexpected complications during a short stay at a Los Angeles hospital for an undisclosed illness, his publicist Dick Guttman said.
His best supporting actor Oscar win for his role in Tim Burton’s 1994 Ed Wood capped a long and versatile career, filled with remarkable ups and downs.
The chameleon-like actor first gained fame as the crafty master of disguise Rollin’ Hand in the 1960s TV show Mission: Impossible.
The series, which also starred Landau’s wife, Barbara Bain, became an instant hit when it debuted in 1966.
But Landau and Bain – who divorced in 1993 after having two daughters – left the show three seasons in due to a salary dispute.
Landau is also remembered for a role he did not take.
He was Star Trek creator Gene Roddenberry’s first pick to portray pointy-eared Mr Spock.
Landau turned it down.
The iconic role eventually went to Leonard Nimoy, who replaced Landau in Mission: Impossible.
After a brief, but impressive Broadway career, New York-born Landau made his film debut in 1959.
He first played a soldier in Pork Chop Hill and then a villain in the Alfred Hitchcock classic North By Northwest.
His film career languished for years and he worked on third-rate projects, such as the 1981 TV movie The Harlem Globetrotters on Gilligan’s Island.
“You know, I’ve always felt, pound for pound, I’m one of the best guys around; but you get stuck in people’s eyes in a certain way, and it takes an imaginative director who will look at you and realise you can play different kinds of parts because you are an actor,” Landau told the New York Times.
His career was revived in 1988 when he received an Oscar nomination for his work in Francis Ford Coppola’s Tucker: The Man and his Dream.
He was nominated again a year later for his role as an adulterous husband in Woody Allen’s Crimes and Misdemeanors.
His third nomination was for Ed Wood, Burton’s affectionate homage to a man widely seen as the worst Hollywood filmmaker of all time.
Landau portrayed washed-up Dracula actor Bela Lugosi in his final years, alongside Johnny Depp, Bill Murray and Sarah Jessica Parker.
After winning his Oscar, Landau gushed: “My God! What a night. What a life. What a moment. What everything!”