Vince Acors was in Dubai on holiday when he met Michelle Palmer
They were fined 1,000 dirhams (£160; $350) and will be deported after serving their sentences.
Their lawyer says the pair will appeal against the verdict.
A spokesman for the judge said Acors and Palmer would be jailed and then deported for the offences of unmarried sex and public indecency.
He added that they had been fined for being drunk in a public place.
"The sentence of three months is usual in these cases. We get many cases of this kind," he said.
Acors and Palmer were not at Dubai's Court of First Instance to hear the ruling but had been ordered not to leave the emirate.
Their defence lawyer Hassan Matter said the pair were upset but not surprised by the verdict.
"They are not guilty but they were prepared for this," he said.
I'm not happy... It's normal for a sentence to be six months to a year for an offence such as this
Faisal Abdelmalek Ahli,
"I think I have a chance in the appeal court. I have 15 days to appeal. I have to find the reason why the judge gave three months."
Acors and Palmer will remain on bail until their appeal is heard at the court.
Senior prosecutor Faisal Abdelmalek Ahli said he was disappointed at the length of sentence.
"I'm not happy," he said, speaking outside court. "It's very light. It's normal for a sentence to be six months to a year for an offence such as this."
Mr Ahli said he expected Acors and Palmer to serve their full three-month term in a Dubai prison.
"Sometimes people serve half their sentence, but this is so short I expect they will serve it all," he said.
'Kissing and hugging'
The pair were arrested on Jumeirah Beach hours after meeting at a champagne brunch at Dubai's five-star Le Meridien hotel.
A police officer told the court he had warned the pair about their inappropriate behaviour, but returned later to find them having sex on a sun lounger.
Palmer, who was sacked from her job in Dubai as a publishing executive after her arrest, said in a statement she and Acors had been "just kissing and hugging".
Mr Matter said witness statements, including one from the police officer, were "wrong" and medical examinations had proved Palmer had not had sex on the beach.
Friends of Palmer say she has been admitted to hospital in recent weeks suffering from anxiety and depression.
The case has turned the spotlight on the lifestyle of the 120,000 British residents of the United Arab Emirates.
The BBC's Christian Fraser, in Dubai, says there have been concerns lately that tourists are ignoring the emirate's strict Islamic laws and that the outcome of this case will be a warning that such drunken behaviour will not be tolerated in public.
Pauline Crowe, chief executive of UK charity Prisoners Abroad, said the case served as a timely reminder to people to be aware of local customs and laws because ignorance would not be accepted as a defence in court.
She said: "As this case illustrates, what may seem like an innocent act or misdemeanour in the UK can often land people in serious trouble when abroad."