Bangladesh’s parliament made its call in April. Prime Minister Sheikh Hasina argued in support of the motion before the General Assembly in September.
Bengali is spoken by more than 250 million people around the world.
The UN has six official languages – English, French, Spanish, Russian, Chinese and Arabic.
English and French are the daily working languages of the organisation, although English is more frequently used than French.
The West Bengal state assembly resolution was adopted unanimously on Monday.
“Thousands have died for the cause of this language. It is our mother tongue and we are proud of it,” West Bengal Chief Minister Buddhadev Bhattacharya said after the resolution was adopted.
The West Bengal government has now asked the Indian government to forward the appeal to the UN. India’s main language, Hindi, is not one of the UN languages.
“The Bangla bhasha (Bengali language ) is spoken by over 250 million people worldwide, primarily in Bangladesh and the Indian state of West Bengal,” Ms Hasina told the UN General Assembly in September.
“Given the rich heritage of Bangla language, and its singular place as a symbol of people’s faith in the power of languages to sustain cultures, and indeed the identity of nations, I seek support of the membership of the UN General Assembly for its acceptance as an official language of the United Nations,” she said.
Sheikh Hasina requested the West Bengal government and Bengali ministers in the Indian cabinet to “push forward” the case for Bengali.
Support for the plea to declare Bengali as an official language of the UN has come from the Indian states of Assam and Tripura, both of which have a sizeable Bengali-speaking community.
While not widely spoken outside the region, Bengali is the language of famed poet Rabindranath Tagore, who won the Nobel Prize for literature in 1913.
Bangladesh observes a language martyrs day on 21 February to remember those who fell to Pakistani bullets on that day in 1952 as they pushed for recognition of Bengali as an official language of Pakistan.
Bengalis in Assam and north-east India observe 19 May as their language martyrs day to remember the 11 Bengalis who were killed in police firing in the southern Assamese town of Silchar while fighting for their language rights.