Peru’s Congress chose lawmaker Francisco Sagasti as caretaker president in hopes of ending a deepening political crisis that had resulted in deadly protests and a power vacuum.

Mr. Sagasti will become the country’s third president in a week after Congress voted to impeach Martín Vizcarra last Monday. His successor, Manuel Merino, resigned on Sunday, just six days after taking office, after large protests over the impeachment led to the deaths of two young men following a police crackdown.

Mr....