The group of young men broke down the door of the office and "started smashing everything", forcing staff to escape through windows, the prominent rights group Committee Against Torture said on its Twitter account.
The young men also attacked the office's car with sledgehammers, according to the group.
The NGO said its staff had phoned police and investigators but they did not immediately react to the attacks, which came after a crowd of protesters gathered outside with placards attacking the group's head, Igor Kalyapin.
The Committee Against Torture's office was also torched in December last year in retaliation after it criticised Chechen strongman Ramzan Kadyrov for urging collective punishment of the families of Islamist insurgents.
Activists from the group appeared in a documentary recently released by an organisation run by Mikhail Khodorkovsky, a former oligarch and Kremlin foe, alleging abuses and corruption by Kadyrov.
The right group posted CCTV footage on Wednesday of the attackers using a blowtorch to open the door of a flat used by activists.
Grozny state television reported that members of the public had gathered for a "spontaneous picket" of the group's office and "expressed their protest against the politicisation of rights activism".
Protesters highlighted the "double standards" of the rights group and said its activities were carried out "to order", Grozny television said, implying those orders were from the West.
Putin's spokesman Dmitry Peskov said he had not yet informed the president of the incident but would do so "when we get detailed information", the RIA Novosti state news agency reported.
