The resolution, moved by state Parliamentary Affairs minister Partha Chatterjee in a special session, was backed by members of the Congress and Left Front while the BJP opposed it.
The Left and the Congress members, however, expressed their dismay at the state government not allowing them to move a resolution against the controversial Citizenship Amendment Act earlier this month and wanted to move some amendments.
However, they did not press ahead with their amendments following appeals from Chief Minister Mamata Banerjee and Chatterjee.
Banerjee, in her speech, described the resolution as one upholding humanity, which the law flouts.
"This is no Hindu-Muslim issue. It is a matter of humanity. This law is a shame for humanity," she said.
Banerjee said the CAA, National Population Register (NPR) and National Register of Citizens (NRC) are inter-connected.
"These are the same thing. So this is not the time for us to quarrel. We all have to fight against dictatorship," she said, in an appeal to the Congress-LF members.
The resolution was passed under rule 169 of the rules of business of the House.
It was adopted without voting, making West Bengal the fourth state in the country to pass such a resolution in its legislature.
The CPI-M-led Left Democratic Front (LDF) government in Kerala and Congress-ruled Punjab and Rajasthan have already passed similar resolutions in the respective Assemblies demanding scrapping of the CAA.
Telangana Chief Minister K. Chandrashekhar Rao has also announced that the state Assembly will pass a resolution against the Act.
Leader of the Opposition Abdul Mannan, of the Congress, and the Left Front legislature party leader Sujon Chakraborty, of the CPI-M, were present in the House.
BJP legislator Swadhin Sarkar opposed the resolution.
On January 20, Chief Minister Banerjee announced that a resolution against the Citizenship Amendment Act will be passed in the state Assembly.
Banerjee and her Trinamool Congress have been at the forefront of the widespread protests against the legislation in the state, which also saw violence and vandalism after the law was passed last month.
The Congress and Left had earlier in the month tried to bring such a resolution in the Assembly but the Trinamool did not agree, and the matter was not included in the list of business of the House.
Since then, these opposition outfits had been ridiculing the Trinamool, describing its reluctance as another evidence of a covert alliance between Banerjee and Prime Minister Narendra Modi.
However, after Banerjee's announcement, Chatterjee wrote to Mannan and Chakraborty seeking their support for the resolution.
The CAA, passed in Parliament last month, seeks to provide Indian nationality to Hindus, Christ, Sikhs, Parsis, Jains and Buddhists who fled religious persecution in Pakistan, Afghanistan and Bangladesh till December 31, 2014.
As per the Act, such communities will not be treated as illegal immigrants now and will be given Indian citizenship.
( With inputs from IANS )