Wlodzimierz Umaniec, also known as Vladimir Umanets, was arrested after visitors discovered a scrawl across the bottom of Rothko's "Black on Maroon" on Oct 7.
The 26-year-old later said he had written the words "a potential piece of yellowism" on the abstract painting to draw attention to Yellowism, an artistic movement he co-founded.
He pleaded guilty to criminal damage over 5,000 pounds ($8,000). Prosecution lawyer Gregor McKinley said restoring the painting would cost around 200,000 pounds ($320,000) and take up to 20 months.
"Complications to this work include the unique painting technique used by the artist and the fact the ink used by Mr. Umaniec has permeated the paint layers and the canvas itself," he said.
Passing sentence at Inner London Crown Court, judge Roger Chapple said it was "wholly and utterly unacceptable" to promote the movement by damaging a work of art that had been "a gift to the nation."
Russian-born Rothko, who died in 1970, was a leading figure in American abstract painting, renowned for large-scale works featuring bold blocks of color.
The defaced painting was one of a series intended to decorate the Four Seasons restaurant in New York. Rothko changed his mind about the commission and gave the works to galleries, including the Tate.
McKinley said auctioneer Sotheby's had valued "Black on Maroon" at between 5 million pounds and 9 million pounds ($8 million and $14 million).
This is not the first time an artwork at Tate Modern has been interfered with. In 2000, two Chinese performance artists attempted to urinate on Marcel Duchamp's urinal sculpture "Fountain."