With about 94.9 per cent of votes tallied, Sanchez rose to 50.10 per cent, while Fujimori dropped to 49.90 per cent.
Most voters who cast ballots in the election on Sunday said they were concerned about crime and the economy, and a rightward shift similar that to seen in other recent Latin American contests had been widely predicted.
Fujimori led early results and exit polls, but Sanchez has been gaining ground since Sunday night as ballots from Peru's rural regions are counted.
Sanchez, a minister under jailed ex-President Pedro Castillo, has aimed to emulate Castillo's rural appeal, even wearing the former leader's wide-brimmed cowboy hat throughout the campaign.
Aside from reforming the constitution, Sanchez has proposed pardoning Castillo - serving an 11-year prison sentence after attempting to dissolve Congress in 2022 - imposing windfall taxes, a wealth tax and reforming mining concessions.
The proposals have drawn support from Peru's vast rural regions, including the country's increasingly powerful informal mining sector, but have unsettled investors.
Fujimori, appearing in her fourth presidential runoff, called for patience as she watched an early lead dwindle.
"We're going to wait until the last (vote) and that's what I hope all Peruvians do," Fujimori said outside her home in Lima.
Several international polling stations have yet to be counted that are expected to favour Fujimori.
Sanchez, speaking to reporters at Congress, said he was "confident and optimistic, but we'll wait for 100 per cent of the vote. What comes next is working for the country because Peru's political instability needs to stop."
Peru's ONPE electoral authority said a full count was expected to be completed by July.
Fujimori had previously distanced herself from her father, Alberto Fujimori, Peru's authoritarian, tough-on-crime former president who was later jailed for human rights abuses. But with voters overwhelmingly concerned with crime this election, Fujimori has since leaned into his legacy.
The winner of the election will become Peru's ninth president in the past decade, and Congress has removed three presidents in the past five years.
