A notice of motion, proposed by Morgan Jones ward councillor Jamie Pritchard, received cross-party support at the meeting on Tuesday, September 17.
The notice of motion requests that the council writes to the Welsh Government to urge a ban on giving animals as prizes on public and private land across Wales.
The motion will be considered at a full council meeting in the near future.
Cllr Pritchard said: “Animal ownership is a big responsibility and is one that should be planned and well thought out.
“Sadly, animals often do not have their welfare needs met prior to, during or after being given as a prize.
“As a council, we have the opportunity to ban the giving of live animals as prizes on council-owned land and ensure the welfare of these animals is not compromised, as well as raising public awareness of the issue.”
Cllr Pritchard added: “I’m delighted that members of the environment and sustainability scrutiny committee endorsed the notice of motion that would see us leading the way on ending this outdated practice. I look forward to it being considered by full council in due course.”
Since 2014, there have been 24 cases reported to the RSPCA of live animals being given as prizes in Wales.
Of these, 60% were goldfish being given at fairgrounds, but other animals including dogs, ducklings and horses were also reported.
Currently it is an offence to give an animal as a prize to anyone under 16, except within a family context, but RSPCA Cymru says this does not go far enough.
Chris O’Brien, from RSPCA Cymru, told a meeting of the council’s scrutiny committee on Tuesday that the authority has an opportunity to “spearhead” the campaign in Wales.
Mr O’Brien said: “It becomes almost a domino effect where other local authorities step up and take action.”
Campaigner Holly Homer, founder of the Operation Goldfish campaign, said she hopes the council’s decision will pave the way for other councils to act on the issue.
Ms Homer said: “This will not only save animals’ lives but raises awareness of the fact that a pet isn’t something you win on impulse but instead a creature that you are legally required to care for properly throughout their entire life.
“It is a pleasure to work alongside RSPCA Cymru & Cllr Pritchard. Together, I hope, we will put a stop to these archaid traditions.”
Additional reporting from the Local Democracy Service