The mid-century plane, a Junkers JU-52, crashed in the Piz Segnas mountain range, near the Austrian border.
"The police have the sad certainty that the 20 passengers perished", said a spokesperson for regional police in the canton of Grisons.
Police have given few details but have confirmed the crash, saying the aircraft came down on a mountainside about 2,540m (8,333 ft) above sea level.
Police said they were not aware of any distress call from the aircraft before it crashed.
Among those killed were 17 people from Switzerland and a three-person family from Austria.
Knecht also said the 79-year-old plane was most likely not equipped with crash-resistant data recorders that more modern aircraft have.
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Almost 5,000 Ju-52 planes, a product of Germany's Junkers, were manufactured between 1932 and 1952.
The wreckage was discovered engulfed in flames, and the cause of the crash is now being investigated by the Federal Prosecutor's Office, in cooperation with local Swiss authorities.
The area, which includes a glacier, is popular with hikers.
The Junker JU52 HB-HOT aircraft, built in Germany in 1939 and now a collectors item, belongs to JU-Air, a company with links to the Swiss air force, the ATS news agency reported.
The plane did not catch on fire before or after it hit the mountain, and investigators have not found any signs the aircraft lost parts or broke up in the air before the crash, Knecht said. Its pilots are ex-military and professional pilots, all of them volunteers. The airline suspended flights until further notice.
Based out of Duebendorf in canton Zurich, JU-AIR offers sightseeing and adventure flights around the Alps.