Estonian authorities claimed that monuments that encourage social divisions must be removed, and the sentiment gained added momentum following clashes in front of the "Bronze Soldier" monument last year between Estonian nationalists and Russians living in the country.
The bill, which was passed with 46 deputies voting for and 44 against its adoption, will now be sent to the president for signing.
In line with the approved bill, the "Bronze Soldier" monument is now a prohibited construction, and the Estonian authorities are obliged to demolish it within 30 days of the law's entry into force.
In late January, Russian Foreign Minister Sergei Lavrov said Moscow's reaction to the Estonian law on war memorials would be tough, but he ruled out sanctions against the Baltic state.
"Our reaction should be strong and tough, aiming to mobilize public opinion, above all in the European countries, to prevent the desecration of the memory of those who fought against Nazism. But I would not go for sanctions," Lavrov said.
Earlier, on January 10, Estonia's parliament approved a law permitting the reburial of the remains of Soviet soldiers who died fighting German invaders during the Second World War, but who are seen by many Estonians as having been occupiers.
Russia has repeatedly drawn the European Union's attention to attempts by Estonia, which declared its independence from the Soviet Union in the early 1990s and joined NATO and the EU in 2004, to glorify Nazi Germany, including with parades by former Nazi SS fighters.
Moscow has also harshly criticized Estonia's discriminatory policies with respect to ethnic Russians who moved to the republic following its annexation by the Soviet Union in 1940, and their descendents.
Many members of Estonia's Russian community are denied citizenship and employment rights, and cannot receive an education in their native language.
The human rights group Amnesty International has condemned the situation in the Baltic country, and called on its leadership to respect the rights of ethnic Russians.