Professor Ambikesh Mahapatra teaches chemistry at Jadavpur University. A court released him on bail.
Police say he forwarded doctored images of Ms Banerjee, former Railway Minister Dinesh Trivedi, whom she recently sacked, and his replacement Mukul Roy.
The cartoon went viral on the social networking site Facebook.
Ms Banerjee sacked Mr Trivedi in March for not consulting her before presenting the railway budget, which included small but controversial fare increases.
The cartoon is a take on Oscar-winning late Bengali filmmaker Satyajit Ray's film Sonar Kella, or The Golden Fort.
It criticises Ms Banerjee's performance as a former railway minister. It also pokes fun at her for sacking Mr Trivedi from the same role.
Prof Mahapatra was arrested for "spreading derogatory messages against respectable people", said Sujoy Chanda, Deputy Commissioner of Calcutta police.
He did not say who these "respectable people" were.
The professor has been charged under India's information technology (IT) Act.
Prof Mahapatra was also attacked on Thursday night, allegedly by supporters of Ms Banerjee's Trinamool Congress [TMC] party.
His fellow professors and intellectuals have condemned his arrest, calling it an attack on free speech.
The chief minister, who came to power in 2011 bringing to an end the state's 34-year-old Communist rule, has recently attracted a lot of negative publicity.
The government's recent order to paint Calcutta blue was criticised by many.
Many of Ms Banerjee's high-profile supporters, including party MP and singer Kabir Suman, theatre actor Kaushik Sen and educator Sunanda Sanyal, have also begun criticising her government.
Indian police have arrested a professor in Calcutta for allegedly posting on the internet cartoons ridiculing West Bengal Chief Minister Mamata Banerjee.