Anderson plays the role of Elwood P. Dowd, an amiable but eccentric man whose best friend is an invisible, 6 ft 3+1⁄2 in-tall (1.92 m) white rabbit named Harvey. As described by Elwood, Harvey is a Púca (pooka), a benign but mischievous creature from Celtic mythology. Elwood spends most of his time taking Harvey around town, drinking and talking at various bars and introducing Harvey to almost every person he meets, much to the puzzlement of strangers, though Elwood's family and friends have accepted Harvey's (supposed) existence.
His older sister Veta (Kurtz) and his niece Myrtle Mae live with him in his large estate, but have become social outcasts along with Elwood due to his bizarre behaviors in public and his obsession with Harvey.
After Elwood ruins a party Veta and Myrtle Mae had arranged in secret, Veta finally tries to have him committed to a local sanatorium. In exasperation she admits to the attending psychiatrist, Dr. Sanderson, that she sees Harvey every once in a while herself and that she has started to believe in his existence. Mistaking Veta as the real mental case, Sanderson has Elwood released and Veta locked up. Dr. Chumley, head of the sanatorium, discovers the mistake and realizes he must bring Elwood back, searching the town with Wilson, an orderly. With Veta's help, Chumley eventually tracks Elwood to his favorite bar, "Charlie's", and decides to confront him alone.
Four hours later, Wilson returns to the sanatorium, but learns from Sanderson and nurse Kelly that Chumley has not returned with Elwood. They arrive at Charlie's and find Elwood alone; he explains that Dr. Chumley had wandered off chatting with Harvey after several rounds of drinks. When asked, Elwood explains that he met Harvey down a street beside a lampost one day, leaning against a pole. Elwood was surprised when Harvey called him by his name and soon they started taking about various things in the world and became fast friends. They had since enjoyed going to bars and socializing with other people. Convinced Elwood is insane and may have harmed Chumley, Wilson calls the police and has Elwood escorted back to the sanatorium.
Chumley returns to the sanatorium disheveled and paranoid, and is followed by an invisible presence. When the others arrive, Chumley invites Elwood to his office. In private, Chumley says that he now knows Harvey is real, and Elwood explains Harvey's various powers, including his ability to stop time, send anyone to any destination for as long as they like, and then bring them back without a minute passing. Veta arrives with Judge Gaffney and Myrtle Mae, prepared to commit Elwood to make him "normal". They are convinced by Sanderson that an injection of a serum called Formula 977 will stop Elwood from "seeing the rabbit".
As they prepare the injection, Veta tries to pay the cab driver but, emptying her purse, is unable to find her smaller coin purse. She interrupts the injection procedure and asks Elwood to pay the driver. Elwood while paying the driver starts a conversation about the driver's occupation. Warmed by Elwood's kindness, the cab driver explains how he has driven many people to the sanatorium to receive the same medicine and how they enjoy the ride here, watching the sunset and talking about various things but on the way back, the same people start shouting at the cab driver, complaining how busy they are and that they are always running short of time. He warns Veta that Elwood will become like them, "a perfectly normal human being,...and you know what they are." Veta is upset by this, and halts the injection, bringing Elwood out of the examining room. At this, Judge Gaffney becomes annoyed and leaves. She then finds her coin purse, and realizes that Harvey had intervened to save her brother.
As Elwood is about to leave, he finds Harvey near a staircase; they start walking back, but Dr. Chumley finds them and asks Elwood where he can find his own púca, to which Elwood replies, "He'll find you...When you are ready."
In the ending scene, Veta and Elwood are seen smiling and playing the piano together in their mansion while an empty chair rocks back and forth, supposedly with Harvey sitting on it.
