Other constitutional changes expand parliament's role, but they also strengthen the already-powerful role of the president.
The president will have the right to dissolve parliament if it refuses to support the candidacy of a minister proposed by the head of state three times in a row.
They will also have a greater say over the work of the Constitutional and Supreme Courts and prosecutors.
The reform also strengthens the role of the State Council, currently an advisory body.
Under the current rules, 67-year-old Putin's current term in the Kremlin would expire in 2024.
Putin has cultivated a reputation as a guarantor of the Russian state's stability, in contrast to the turbulence of the post-Soviet 1990s before he came to power.
Although his approval rating has declined to a personal record low as Russia struggles with the coronavirus pandemic, Putin still maintains broad support.
About 59% of Russian adults approve of Putin's work as president, according to a nationwide survey last month by the country's largest independent pollster, Levada Centre.
Opposition leader Alexei Navalny has slammed the vote as a populist ploy designed to give Putin the right to be "president for life".
"It is a violation of the constitution, a coup," he said this month on social media.
Two regions with large numbers of voters - Moscow and Nizhny Novgorod - allowed electronic balloting amid the coronavirus pandemic.
Monitoring the vote became more challenging due to hygiene requirements and more arcane rules for election observers. Kremlin critics argued that these would increase opportunities for vote fraud.
Russia's weakened and fragmented opposition split into two camps over the amendments: those who called for a boycott of the vote, like Navalny, the most visible Kremlin foe, and those who advocated voting against the constitutional changes.
Most observers expect the Kremlin to get its way, regardless of the opposition's strategies.
"People are angry at the government, but they still don't have any alternative to Putin," Gleb Pavlovsky, a political analyst and former Kremlin political consultant, told the Associated Press news agency.
Moscow-based political analyst Ekaterina Schulmann warned that the balloting will likely fail to serve its designated purpose of cementing Putin's rule as the economic pain from the coronavirus deepens.
"I think the vote will not be perceived as the legitimising one," she said.