Rossello, who had on Sunday insisted he would stay on, said he would continue working until August 2 to allow an orderly transition, the BBC reported.
"I announce that I will be resigning from the Governor's post effective Friday, August 2 at 5 p.m.," Rossello said in a video statement on Wednesday evening.
His announcement triggered celebrations on the streets of the capital San Juan. Rossello will be replaced by the US Caribbean island territory's Justice Secretary Wanda Vazquez.
The Governor has been at the centre of a group text message scandal that has already led two top officials to resign. The leaked messages revealed sexist, profane and homophobic comments.
The chat, which contained 880 pages of exchanges between the Governor and 11 male allies, was leaked on July 13 and led to days of protests outside the Governor's mansion here. The chat has messages in which Rossello and several members of his inner circle in the government mocked and insulted journalists, political leaders of all colours, social collectives and well-known personalities.
On Monday, thousands of people took to the streets, demanding Rossello's immediate resignation. The 40-year-old Governor is leader of the New Progressive Party, a Puerto Rican political party.
On Sunday, in an attempt to appease protesters, he said he would not seek re-election next year. "I hear you," Rossello said in a Facebook video. "I have made mistakes and I have apologised."
"I know that apologising is not enough," he added. "A significant sector of the population has been protesting for days. I am aware of the dissatisfaction and discomfort they feel. Only my work will help restore the trust of these sectors."
At a press conference on Tuesday, the Governor had admitted that his participation and statements in the private chat were improper but not illegal and reiterated that he would remain in his post.
( With inputs from IANS )