Asrani was born in a middle class Sindhi family. His father migrated to Jaipur after the Partition of India and opened a shop selling carpets. He has four sisters and three brothers: two elder and one younger. Asrani was highly uninterested in business and very weak in mathematics. He completed his matriculation from St. Xaviers School and did his graduation from Rajasthan College, Jaipur. He simultaneously worked as a voice artist on All India Radio, Jaipur, to pay for his education. His ultimate aim was to become an actor.
He began to learn acting from Sahitya Kalbhai Thakkar from 1960 to 1962. In 1962, he traveled to Bombay and began to seek opportunities to act. In 1963, Hrishikesh Mukherjee advised him to learn acting professionally. In 1964 Asrani joined the Film Institute in Poona (now Film and Television Institute of India, Pune) and finished his course in 1966. He impressed many others and made his acting break in 1967 with a Gujrati movie opposite budding actress Waheeda (not Waheeda Rahman, the Hindi actress). He acted in four other movies, mainly as actor or supporting actor in Gujrati from 1967 to 1969. His debut film in Hindi was Hare Kaanch Ki Chudiyaan where he appeared as Biswajeet's friend. He acted in Khoon Pasina, marking a serious role other than comedy. It was from 1971 that he started getting more offers as the main comedian in films or as the close friend of the lead actor. He did 96 movies from 1972-79. Directors like Hrishikesh Mukherjee, Atma Ram, Gulzar repeatedly cast him in the period 1971-1974 and his work got noticed through these films as they were hits. These roles he did from 1971-1974 set the tone for a number of supporting and comic roles in his career and in 70's his demand was at its peak as he appeared in 101 films from 1970-1979 and had effective roles in them. He was hugely talked about in the 1970s and is one of Rajesh Khanna's close friends. He worked with Rajesh Khanna in 25 films which includes 21 hit films.