He said he had done so at the direction of "the candidate", for the "principal purpose of influencing [the] election".
The admission was related to hush money paid to Mr Trump's alleged lovers.
Mr Trump has not commented. In May, he admitted reimbursing Cohen for paying one of the women, having earlier denied any knowledge of it.
Mr Cohen, 51, admitted eight counts on Tuesday, including tax and bank fraud, in a plea deal with prosecutors which may see his prison sentence reduced from 65 years to five years and three months.
On the same day, a jury in Alexandria, Virginia, convicted former Trump campaign chairman Paul Manafort of bank and tax fraud charges.
The developments in the Cohen and Manafort cases have revived speculation that the president himself may face legal proceedings.
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