All the action takes place on Alcina's enchanted island.
The background of the opera comes from the poem Orlando Furioso. The heroic knight Ruggiero is destined to a short but glorious life, and a benevolent magician is always whisking him away from the arms of his fiancé, Bradamante. Bradamante spends vast portions of the poem chasing after him. Just before the opera begins she has rescued him from an enchanted castle, only to have her flying horse (a hippogriff) take a fancy to Ruggiero and fly off with him. They land on an island in the middle of the ocean. As the hippogriff begins to eat the leaves of a myrtle bush, Ruggiero is startled to hear the bush reveal that it was once a living soul named Sir Astolfo.
The island belongs to the sister sorceresses Alcina and Morgana. The beautiful Alcina seduces every knight that lands on her isle, but soon tires of them and makes them into stones, animals, plants, or anything that strikes her fancy. Ruggiero strides off to meet this sorceress—and falls under her spell.
Scene 1. A deserted spot surrounded by hills
Bradamante, disguised as her brother Ricciardo, and her tutor Melisso have been ship-wrecked and cast ashore while searching for her lover Ruggiero. They meet Morgana, Alcina's sister, who falls in love with the disguised Bradamante and promises to take them to Alcina. With a clap of thunder the hills part and Alcina is revealed, surrounded by her court, including Ruggiero and a young boy called Oberto. She receives the travellers graciously and tells Ruggiero to show them the sights of the island.
Bradamante confronts Ruggiero, who recognises her as Ricciardo but denies any interest in Ricciardo's sister - he is the faithful lover of Alcina.
He runs off to find her and Oberto asks the travellers if they have seen his father Astolfo. They too had been cast ashore by a storm, but his father, after a welcome from Alcina, had disappeared. Bradamante suspects that Astolfo has been changed into a wild beast, like Alcina's other victims.
Oronte, Alcina's commander-in-chief, rightly suspecting the constancy of Morgana, whom he loves, charges Bradamante with having stolen Morgana's love. Morgana defends Bradamante and insults Oronte. Bradamante tries to calm their mutual recriminations.
Scene 2. A room in Alcina's palace
Oronte, coming upon Ruggiero who is sighing for the absent Alcina, decides to alleviate his own jealousy by making Ruggiero jealous too, and concocts a tale that Alcina now loves "Ricciardo" and will no doubt soon add Ruggiero to her collection of discarded and transformed lovers. Ruggiero believes him and heaps reproaches on the puzzled Alcina, who assures him that her feelings are unchanged.
Bradamante accuses Ruggiero of disloyalty, but he retaliates with defiance, accusing her, as Ricciardo, of having stolen Alcina's love. Bradamante discloses her identity, but Melisso, worried that Ruggiero is not yet ready for this information, convinces him that she is not really Bradamante.
Morgana warns Bradamante that Ruggiero has persuaded Alcina to change her into a wild beast, so Bradamante tells Morgana to assure Ruggiero that she does not love Alcina, but another. Morgana departs happily, thinking she is that other.
Alcina laments Ruggiero's jealousy, hoping that their love will soon be as untroubled as before.
Scene 1. A room in Alcina's palace
Melisso, disguised as Ruggiero's tutor Atlante, reproaches him with having abandoned the path of glory and gives him a magic ring which brings him to his senses.
Ruggiero regrets his faithlessness to Bradamante and wishes to send a message of defiance to Alcina, but Melisso advises him to pretend that he still loves her and make his escape on the pretext of going hunting. Bradamante again reveals her identity, only to have Ruggiero reject this revelation as another of Alcina's deceptions.
Scene 2. Near the palace gardens
Alcina is preparing to change Ricciardo into a wild beast to appease Ruggiero, while Morgana tries to dissuade her and Ruggiero assures her that he is no longer jealous, so drastic measures are no longer necessary. Alcina notices that Ruggiero is not in his usual spirits and he suggests a hunt as a restorative. She consents and he departs. Oberto continues to lament his father's disappearance and Alcina falsely raises his hopes of a speedy reunion.
Oronte brings the news that Ruggiero is planning to flee and Alcina prepares to foil this plan. Oronte tells Morgana that her new love is about to leave her, but she refuses to believe this and departs scornfully, leaving him to lament her power over him.
Ruggiero is at last convinced that Bradamante is really herself. Morgana discovers them embracing, and, apparently taking in her stride the revelation that Bradamante is a woman, reproaches her with being a faithless guest and Ruggiero with betraying Alcina. Ruggiero looks forward to the ending of the enchantments.
Scene 3. An underground magic chamber
Alcina begins to make spells to bind Ruggiero to her, but loses heart and casts her wand aside.
Scene 1. The entrance to the palace
Morgana tries to ingratiate herself with Oronte, who pretends indifference, but has to admit to himself that he still loves her. Alcina upbraids Ruggiero for trying to leave her. He tells her that his betrothed Bradamante now has his love, and she threatens vengeance, though unable to obliterate her tender feelings for him. Bradamante and Melisso join Ruggiero to plan their campaign. Melisso tells them that the island is surrrounded by Alcina's enchanted monsters and advises Ruggiero to take the Gorgonian shield and the winged horse (items not previously mentioned) to help him in the fray. Although worried at leaving his beloved, Ruggiero sets off, followed by Melisso and Bradamante, who vows to free those lying under enchantment. Oronte announces to Alcina the complete defeat of her forces and she laments her cruel fate.
Scene 2 Outside the palace
Alcina tries to make Oberto kill a lion, but he realises that it is his father, and threatens to turn the spear on Alcina herself. Ruggiero and Bradmante confront Alcina, each advising the other not to be taken in by her deceptions. She tries both pleading and threats in vain.
Ruggiero returns the defeated Oronte's sword and then smashes the urn which holds Alcina's secret power. All her spells are broken and the rocks, trees and animals, including Astolfo, Oberto's father, resume their human shapes. They rejoice at their liberty and all celebrate the triumph of love.