Passengers who refuse to tell authorities where they are quarantining for two weeks after arriving in Ireland face six months in prison or a €2,500 fine.
The Cabinet agreed to make it mandatory for everyone arriving in Ireland to fill in a passenger location form outlining where they will be self-isolating for a fortnight after their arrival.
The new regulations will come into force next week, on Thursday, and be reviewed on June 18.
The regulations mean gardaí will be able to check up on passengers who filled out the form.
Gardaí can press charges against a person who is found not to be staying at the address they listed.
It will be an offence to refuse to fill out the form or to enter false information.
Breaching the regulations is punishable by a fine of up to €2,500 or six months in prison.
Passengers transiting to another jurisdiction, certified international transport workers, air and maritime pilot/masters and crew, will not have to complete the form.
Individuals from Northern Ireland will only have to fill out a portion of the form.
Health Minister Simon Harris said the new measures are "extraordinary", but "necessary in a time of a public health crisis".
"We continue to advise everyone against non-essential travel. However, if a person does arrive into Ireland, they will be legally obliged to fill out this form, regardless of their nationality.
"The form will be used to facilitate a system of follow-up checks to make sure people who travel to the country are staying where they said that they would. The form will also ensure more accurate and quicker contact tracing, should we have a confirmed case on a flight or ferry coming into Ireland.
"Every measure we take is aimed at stopping the spread of Covid-19 and protecting people from this virus. This is no different," he added.
Meanwhile, Sinn Féin TD Darren O'Rourke said every passenger arriving in Ireland should have their temperature checked on arrival.
"Temperature screening can also act as a deterrent for those thinking of travelling who may be unwell and provide a level of reassurance to other members of the travelling public," Mr O'Rourke said.
"Temperature screening can also act as a deterrent for those thinking of travelling who may be unwell and provide a level of reassurance to other members of the travelling public," he added.