It is a "desperate attempt" of the opposition to spread fear among Muslims, he alleged.
Sawant said Indian citizens belonging to any religion need not worry either about the Citizenship Amendment Act (CAA) or the National Register of Citizens (NRC).
BJP working president J P Nadda would be addressing a rally here on January 3 in support of new citizenship law.
Various parties, including the Congress and NCP, have been opposing the CAA, which seeks to grant citizenship to non-Muslim refugees who came to India before December 31, 2014, to escape religious persecution in Pakistan, Bangladesh and Afghanistan.
Hitting out at those against the Act, Sawant, in a series of tweets, said, "The opposition is spreading illusions for vote bank politics, it is being propagated amongst Muslims that CAA is a law to take away their citizenship. This is nothing but a desperate attempt by the opposition to spread fear among Muslims."
The CAA has been enacted through a democratic process and gives citizenship to refugees who have had "centuries of cultural affinity with Indian ethos", he further tweeted.
"Indian citizens belonging to any religion need not worry at all, either about CAA or NRC," he said.
"The CAA does not change any existing legal provision which allows any foreign national to apply for Indian Citizenship through registration/naturalization," the chief minister said in another tweet.
He said the new law "empowers persecuted minorities" in Pakistan, Afghanisthan and Bangladesh by giving Indian citizenship. "It applies to those who have migrated before December 31, 2014, on establishing their residency in India for five years."
"CAA categorically addresses concerns of minorities of Pakistan, Afghanisthan and Bangladesh. Members of the majority community in these countries who want to apply for Indian Citizenship will still be able to, as per the existing provisions of Indian Citizenship," Sawant added.
The BJP has been holding awareness campaigns in different Assembly segments of Goa in support of CAA.