After a free vote Members also agreed to write to the Government calling for an outright ban on the award of live animals as prizes on all public and private land.
Cllr Louise Clack (Lab) raised concern about the number of cases reported to the RSPCA regarding animals given as prizes at fairs, via social media and through other channels.
Her motion was seconded by the Council and Conservative Group Leader, Cllr Kelham Cooke, while the Independent Group leader Cllr Paul Wood and Cllr Amanda Wheeler (Lib Dem) also gave their support.
The issue, along with an RSPCA campaign associated with it, was brought to Cllr Clack's attention by a resident of her Earlesfield constituency in Grantham.
"My first reaction was that I didn't think things like this were still legal," said Cllr Clack.
The RSPCA is asking councils across the country to ban the awarding of live animals as prizes on their land and for them to join calls for a nationwide ban.
Cllr Cooke said: "I am delighted to support this important stance on the issue of animal welfare. Pets should be brought in as part of the family after careful consideration of the care and attention they will need, not simply after playing a fairground game.
"Many of us enjoy a visit to the fair or similar events, but while we are doing so animals should not be suffering and this motion means that can no longer happen on South Kesteven District Council land."
Cllr Clack said while the majority of live animals awarded as prizes were goldfish, the RSPCA had verified reports of other animals, including ducklings, dogs and horses also being awarded as prizes at events around the country.
"Animals are being transported and held in conditions that do not meet their specific needs," she said. "This often results in suffering and death before the animals can be awarded as prizes or before the prize winner can get them home.
"If we focus on the example of goldfish, we can see that they are very easily stressed and often suffer shock, oxygen starvation or death caused by sudden temperature change in water.
"Any Members who have kept fish as pets know that you need a tank, filter and pump commensurate with the requirements of the species. Families who want to have fish as pets can get them, and the required kit, from a reputable supplier that can also provide the necessary advice and support for pet ownership.
"SKDC does not have the power to change central legislation and prevent this cruel and unnecessary practice nationally. However, it does have the power to ban this practice on its land and thereby send a message that the safeguarding of animal welfare is a priority for us."
Under current legislation in England it is only an offence to give away an animal as a prize if the recipient is under 16 and not accompanied by an adult.