Kvasha graduated from the Moscow Art Theater School, where he was taught by Aleksander Karev.

After graduation, he joined the Moscow Art Theater troupe and performed there for two years (1955-1957).

In his last years, Kvasha hosted the TV talk show Wait for Me (Russian: Жди меня) on Russia's Channel One.

In the spring of 2001, Kvasha spoke in support of the journalistic team of the NTV channel and criticized the actions of President Vladimir Putin.

[3] In October 2008, he signed an open letter of appeal in defense and support of the release of Yukos Oil Company lawyer Svetlana Bakhmina.