His publicist Dick Guttman confirmed the death, saying: "We are overcome with sadness."
Landau won a Best Supporting Actor Oscar in 1995 for portraying the horror movie star Bela Lugosi in Ed Wood.
He died on Saturday in Los Angeles of "unexpected complications" following a hospital visit.
Landau was born in New York and started out as a cartoonist for the New York Daily News before moving to theatre and then cinema acting.
He featured in the Alfred Hitchcock film North by Northwest and played a commander in Space: 1999 and Geppetto in a live-action version of The Adventures of Pinocchio.
But he turned down the role of Mr Spock in Star Trek, a role that went to his friend Leonard Nimoy instead.
And Nimoy later replaced Landau on Mission: Impossible when the latter left following a dispute over pay.
Many in Hollywood hit social media to pay tribute, including Star Trek actor William Shatner, who played the role of James T. Kirk. PLACE YOUR TEXT ADVERT BELOW:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How I Made Me =N=2.5 MILLION In 7 Days from 2 Common Product I Imported and how you can do the same. Click HERE FOR FREE Details right now Warning To Men: This 3 Foods Is Killing Your Erection. Click Here To Know Them Secret Fruits that increased my Manhood and makes me last 55 minutes+ all time - Click here to get it Warning: Got Arthritis? 3 Simple Ways To Get Cured And Be Permanently Free From Joint Pains. Click Here To See Them REVEALED!! Reduce Your Blood Sugar Level And Reverse Diabetics Within A Short Period Of Time. Click Here To See How MIDNIGHT,Pressed Mr Kilanko Suddenly Couldn't Urinate. Shrink your Enlarged Prostrate Here! Click Here Now