The magazine published photographs of Hollande and Gayet arriving separately at an apartment in January.
The payout was far lower than the 50,000 euros the actress had sought.
The Paris court ordered the magazine to publish the ruling on the front page of its next edition.
Closer caused a political storm in France on 10 January when it published images purporting to show Hollande and Gayet arriving at a Paris flat for alleged trysts.
Although he refused to comment on the allegations, President Hollande announced that he had "put an end" to his "shared life" with long-term partner Valerie Trierweiler shortly afterwards.
During court hearings, Gayet's lawyer said she had been "hunted" by journalists.
"She was assaulted by swarms of photographers… it was like the hunt of a wild animal," he said.
Closer's lawyer maintained that the magazine was justified in published the photographs, saying they were in the public interest because they raised questions about Hollande's "duty of transparency".
Gayet has also filed two criminal complaints: one for a breach of privacy over photographs taken of her inside a car and another accusing paparazzi of "endangering others" while chasing her.
The mother of two, who has acted in more than 70 films over a 20-year career, has kept a low profile since the scandal broke.
During a rare public appearance in New York earlier this month she fended off questions about her relationship with President Hollande, saying: "My private life is my private life."
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