The ban was included as an amendment to animal protection laws and also covers the selling, purchase and possession or dog or cat carcasses.
Taipei had previously banned the sale of dog meat but in 2011 the Taipei Times published claims by activists that slaughterhouses and dog meat restaurants were escaping prosecution.
And a lamb hotpot restaurant was found last year to be serving dog meat to cut costs.
Under the new legislation, anyone found guilty of eating cat or dog meat can be subject to a fine of between NT$50,000 (£1,300) and NT$250,000 (£6,500), according to the China Post newspaper.
Offenders also face public shaming and those who repeatedly break the law face fines of up to NT$5m (£130,000).
The eating of dog and cat meat is not unusual in parts of Asia, despite objections from international animal rights campaigners.
Breeders sell live dogs for meat in and around Yulin, in China’s Guangxi province annually in June.
Residents in Yulin host small gatherings to consume dog meat and lychees in celebration of the summer solstice.
The new Taiwanese law also includes a clause making it illegal to walk a dog by attaching it to a motor vehicle, the China Post said.
One of the leglislators promoting the law, Chiu Chih-wei said there had been several reports of dogs being injured when forced to run alongside a scooter or car driven by their owners.