The Shadow Man, and a Christmas poem called Noel, were found at Our Lady's School, Abingdon.
It is thought Tolkien got to know the school while he was a professor of Anglo-Saxon at Oxford University.
The poems were printed a year before Tolkien's first literary sensation The Hobbit was published.
The Shadow Man is an earlier version of a poem eventually published in 1962 in Tolkien's Adventures of Tom Bombadil collection.
The existence of the poems came to light after American Tolkien scholar Wayne G. Hammond got in touch with the school.
He knew from notes made by Tolkien that two of his poems appeared in a magazine he called The Abingdon Chronicle.
This was actually the 1936 annual of Our Lady's School, which was then run by the Sisters of Mercy.
School principal Stephen Oliver said: "At first we couldn't find the 1936 edition and referred Mr Hammond to the archives of the Sisters of Mercy in London.
"Then, while preparing for an event for former pupils of the school, we uncovered our own copy and I saw the two poems Mr Hammond had been looking for. "
The school hopes to put the annual on display at a future exhibition.

The director of Edward Scissorhands and Beetlejuice threw the switch around 21:30 BST after rapper Professor Green played to the crowds.
The free event took place on the promenade at The Headlands Arena in front of the Tower.
Burton, 58, used Blackpool for scenes in his latest film. "Miss Peregrine's Home for Peculiar Children."
"To be given the town's highest honour is incredible and I'm really excited. I'm sure it will be a wonderful show," he said.
The California-born film director achieved fame directing blockbusters including Corpse Bride, Planet of the Apes, Charlie and the Chocolate Factory and Alice in Wonderland.
Describing himself as Blackpool Pleasure Beach's "biggest fan", Burton chose to shoot scenes in Blackpool for The Killers'  music videos Here with Me and Battle Born.
The Illuminations end on 15 November.
Two poems by author JRR Tolkien have been discovered in a 1936 copy of a school annual.
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Hollywood director Tim Burton has switched on this year's Blackpool Illuminations.