Avargal revolves around the trials and tribulations of a cheerful girl Anupama or Anu (Sujatha), a passionate Bharatanatyam dancer in love with Bharani (Ravi Kumar), an accomplished flutist. Her life changes when her father is transferred to Mumbai on promotion. She slowly loses touch with Bharani, who does not respond to any of her letters, their only mode of communication. When her father falls sick, his colleague Ramanathan (Rajinikanth) supports them. When Ramanathan offers to marry Anu, she is in a fix. She seeks time to decide and writes one last letter to Bharani; when there is no reply, she marries Ramanathan. Soon after, her father passes away.
After marriage, Ramanathan reveals his true colors; very sadistic and possessive, he harasses Anu. He forces her to give up dancing, as the chauvinist in him doesn't allow him to be identified as a dancer's husband. Ramanathan suspects Anu's character and he expresses doubts regarding the paternity of their newborn son. Unable to take any further abuse, and also for the wellbeing of her son, she seeks and obtains a divorce from Ramanathan, refuses alimony, and returns to Chennai to start a new independent life.
In Chennai, Anu gets supportive friends at her office, particularly a widower Janardhan (Kamal Haasan). Janardhan (aka Johnnie), a talented ventriloquist who "talks" through his puppet, Junior Johnnie, goes out of his way to help Anu. When she finds it difficult to find a house, being a divorcee, he offers his own flat and moves to his outhouse. Though he loves Anu, he lacks the courage to tell her. Bharani, Johnnie's neighbor, noticing a new person in the house and a Mona Lisa portrait (which Anu always hangs on the wall), visits and is surprised to see Anu. Anu is surprised too. Anu learns that none of her letters had reached Bharani, since they were intercepted by his mentally-challenged sister. Bharani saw the letters only when his sister was hospitalized. But by then, it was too late as Anu had already married Ramanathan. Unable to forget Anu, Bharani remained single.
When Anu visits an ascetic to discuss her problems and seek his blessings, Ramanathan's estranged mother Leelavathi overhears the conversation and identifies Anu as her daughter-in-law. Leelavathi regrets her son's sadistic behavior; taking advantage of her anonymity, Leelavathi offers to work as a domestic help at Anu's house. Anu renews her relationship with Bharani. Leelavathi also supports Anu's decisions and advises her to move ahead in life. They decide to marry and a date is fixed for a civil wedding.
At the same time, Ramanathan comes to Chennai as her boss — a repentant Ramanathan who now wants to redress the wrongs he had done to her. He goes out of his way to make amends for his mistakes, which confuses Anu. He even celebrates his marriage anniversary and surprises Anu by remembering the death anniversary of her father! He does not reveal that he is Anu's ex-husband to Bharani, who reveals his plans to marry Anu.
Johnnie learns about the love between Anu and Bharani; though disappointed, he continues to support Anu. Leelavathi tries to convince Anu to discard her "Thaali" and go ahead with her marriage to Bharani. But on the marriage day, Anu falls sick and does not turn up at the marriage Registrar's office. Bharani comes to her house, where he is disappointed to see Ramanathan taking care of Anu. It is a piquant situation with three men vying for Anu.
Ramanathan seeks forgiveness and expresses his desire to re-marry Anu. When her colleague Rajathi (Kumari Padmini) visits Anu, she meets Leelavathi and is stunned to see the mother-in-law employed as a maid. Despite Leelavathi's pleas not to reveal her identity, Rajathi, out of frustration, reveals the truth to Anu. Anu is shocked and seeks Leelavathi's forgiveness. She goes to meet Johnnie to seek his advice. That is when Johnnie develops courage and reveals his love for Anu through Junior. Anu expresses her inability to accept his love but seeks his continued support as a true friend. Noticing Anu's predicament and Ramanathan's continued involvement in her life, Bharani decides to move away from her and goes abroad for his musical programs.
Ramanathan's reformed behavior and Leelavathi's sacrifices make Anu agree to become Ramanathan's wife once again. She marries him and reaches his home. Leelavathi there reveals that Ramanathan has not changed even one bit; he had staged this drama to stop her marriage to Bharani. When another lady lands up with her child claiming to be Ramanathan's wife, Anu realizes Ramanathan's hidden agenda. Anu rushes to meet Bharani, but it is too late by then, as Ramanathan has convinced Bharani to marry his rich and ardent fan Gayathri (Kutti Padmini), and the engagement takes place.
Anu confronts Ramanathan and demands the reasons for his vengeance. Ramanathan makes it clear that he could not tolerate his ex-wife living happily; hence he ensured that all her dreams are shattered. Anu resigns her job and leaves the city. Johnnie again comes to her help and makes all arrangements for her travel to Trivandrum, her new destination. When the train leaves the station, Anu is shocked to notice Leelavathi also in the same train taking care of her child. Leelavathi states this is the only way she can find solace for the sins of her son. Anu hugs her and cries for the first time at finding unconditional love and a true companion in her life.
[from the book Pride of Tamil Cinema: 1931 to 2013 by G Dhananjayan, Blue Ocean Publishers]