The laws, which also prohibit the display of Nazi symbols, have passed the state's upper house and are expected to come into effect later this year.
"We strongly condemn any display of hate in our community," Attorney-General Elise Archer said on Wednesday.
"This (law change) is the first of its kind in Australia and will contribute to the creation of a safer and more inclusive Tasmania."
The federal government is moving to ban Nazi symbols, with a proposal introduced to parliament in June. But it doesn't cover the salute.
Victoria is among several states to have banned Nazi symbols and has flagged plans to also outlaw the salute.
Under the law changes in Tasmania, it is an offence to perform a Nazi gesture if a person knows, or ought to know, it is a Nazi gesture.
There is a defence if a person proves the gesture was reasonable and performed in good faith for a genuine academic, artistic, religious, scientific, cultural, educational, legal or law enforcement purpose.
Ms Archer said the law change acknowledged the importance of the swastika to the Buddhist, Hindu and Jain religious communities.
"(It) clearly states that the display of a swastika in this context is not an offence," she said.
"The bill also acknowledges other legitimate public purposes for display, including other religious, cultural, academic and educational purposes."
Mr Archer said police would undergo education and training on the new offences, including on the cultural significance of the swastika.
First offenders face a $3900 fine or three months' jail, with maximum penalties doubling for further offences within six months.
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