Jeenbekov, who has faced calls to step down from protesters and political opponents, said in a statement released by his office that holding onto power wasn't "worth the integrity of our country and harmony in society."
"For me, peace in Kyrgyzstan, the country's integrity, the unity of our people and the calm in the society are above all else," Jeenbekov said.
Kyrgyzstan, a country of 6.5 million people located on the border with China, was plunged into chaos following an Oct. 4 vote that election officials say was swept by pro-government parties. The opposition said the election was tainted by vote-buying and other irregularities.
Protesters then took over government buildings, looting some offices, and the Central Election Commission nullified the election. Opposition then announced plans to oust Jeenbekov and form a new government.
Jeenbekov kept a low profile in the first few days after the vote, using the infighting among protest leaders to dig in. He introduced a state of emergency in the capital, Bishkek, which was endorsed Tuesday by parliament.
Authorities deployed troops to Bishkek over the weekend and introduced the curfew. The move eased tensions in the city, where residents feared looting that accompanied previous uprisings and began forming vigilante groups to protect property. Stores and banks that were closed last week have reopened.
In an effort to stem the unrest, Jeenbekov on Wednesday endorsed the appointment of Sadyr Zhaparov, a former lawmaker who was freed from jail by demonstrators last week, as the country's new prime minister and Zhaparov's new Cabinet.
Zhaparov promised his supporters to push for Jeenbekov's resignation and held talks with the president hours after Jeenbekov signed off on his appointment. After the talks, Jeenbekov said he would stay in the job until the political situation in Kyrgyzstan stabilizes.
But hundreds of Zhaparov's supporters rallied in the capital Wednesday, demanding the president's resignation and threatening to storm his residence. Zhaparov promised on Wednesday he would meet with the president again on Thursday to convince him to step down.
It wasn't immediately clear whether the meeting took place, but the protests demanding Jeenbekov's resignation continued Thursday morning.
Jeenbekov said in his statement that the situation in Bishkek "remains tense" despite the fact that the new Cabinet was appointed the day before, and that he doesn't want to escalate these tensions.
"On one side, there are the protesters, on the other — law enforcement. Military personnel and law enforcement services are obligated to use weapons to protect the State Residence. In this case, blood will be shed, it is inevitable," Jeenbekov's statement said. "I don't want to go down in history as a president who shot at his own citizens and shed blood."
The turmoil marks the third time in 15 years that demonstrators have moved to oust a government in Kyrgyzstan, one of the poorest nations to emerge from the former Soviet Union.
As in the uprisings that ousted presidents in 2005 and 2010, the current protests have been driven by clan rivalries that shape the country's politics.