The Canadian-American singer-songwriter was set to take to the stage as part of her UK tour, but announced shortly after 6pm that she was unable to perform - after doors had already opened.
In a message to her fans, Alanis said: "I feel so sad that I have to cancel tonight's show in Leeds.
"The entire range of my voice is gone today, I thought I could push through like we did last night.
"But it's gone. I love you and I am so sorry and I know I will see you again soon."
Ticket holders took to social media to express their disappointment.
One fan said: "I don't want a refund I want the gig we paid for. Absolutely gutted!!!"
Another added: "I hope she feels better soon, I understand these things happen. However when you’ve been travelling since noon, it’s not the best news!"
First Direct Arena said that ticket holders will be automatically refunded to the method of payment used for purchase.