The new guidelines "epidemiological investigation of a recent group infection case", which led the ministry to consult with South Korea's Center for Disease Control and Prevention on the new rules.
Cats and dogs in South Korea will be tested for Covid-19 if they are exposed to someone with the virus and show suspicious symptoms.
If a pet tests positive for coronavirus its owner must also self-isolate, according to the new regulations.
It comes after after a warning in UK, from experts from the University of East Anglia and research facility the Earlham Institute, that the virus can infect a wide range of animals including cats, dogs, mink and other domesticated species.
The first case of a cat coronavirus infection in the UK was last July.
According to the World Health Organisation, while the disease is mostly spread between humans, there are cases of transmission between people and animals.
Several animals that have been in contact with infected humans, such as minks, dogs, domestic cats, lions and tigers, have tested positive for SARS-CoV-2, the virus that causes Covid-19.
South Korea's animal testing announcements comes just as the country said it would extended social distancing restrictions imposed during its worst coronavirus spread for another two weeks, after new infections re-surged last week.
Despite only having suffered 1,420 Covid-related deaths since the pandemic began in March - compared to more than 100,000 in the UK - there were another 355 confirmed cases in South Korea yesterday, with six new deaths.
The alert for Seoul, where half of the country’s population live, will remain at the second-highest level of 2.5 for the additional two-week period, Prime Minister Chung Sye-kyun confirmed.
A ban on more than four people meeting will also be extended in the run-up to the upcoming Lunar New Year holiday.
The South Korean government confirmed it is continuing to assess the possible "transmission between humans and animals" and said the matter "will be closely monitored and transparently disclosed", urging pet owners not to worry at this stage.
The country's approach to animals during the pandemic appears to be somewhat more light-touch than that of its neighbour North Korea, after the authoritarian state reportedly ordered all pet dogs to be rounded up in August, last year.
Although not explicitly referencing coronavirus, leader Kim Jong-un demanded all pet dogs in the capital Pyongyang were to be be confiscated because they represent Western "decadence".
