The country's imploding economy has resulted in high unemployment and shortages of food and medicine, and hundreds of thousands of people are said to be in need of humanitarian aid.
Prior to Saturday's deadline, Venezuelan citizens wanting to enter Peru only require a national ID card.
Nearly 6,000 Venezuelans entered through the border town of Tumbes on Thursday, officials said, which is almost three times the daily average.
Marianni Luzardo was travelling to Peru's northern border with her two daughters on Friday. "In Venezuela it is almost impossible to get a passport," she told the Associated Press. "We need to get to Peru soon."
Peru's President Martin Vizcarra defended the tougher migration restrictions on Thursday.
"Our country has opened its arms to more than 800,000 Venezuelans," he told reporters at an event in the northern city of Piura. "I think it's completely logical and justified to ask them to bring visas to ensure better control of who enters."
Latin American countries host the vast majority of Venezuelan migrants and refugees. Colombia has the most at 1.3 million, followed by Peru with 768,000, according to UN figures.
What's the situation in Venezuela?
Under the government of Nicolás Maduro, the economy has collapsed and shortages of food and medicines have become widespread.
In parts of the oil-rich country, fuel has become scarce and drivers queue for days at petrol stations. There are also frequent blackouts.