The woman, who does not want to be identified, suffered serious, life-changing injuries while she was delivering mail in Tranent.
She was rushed to hospital for emergency plastic surgery after suffering lacerations to her face.
Despite the horrific incident, the union says the animal is still at large and its owner is yet to be charged with any offence.
Details of the attack emerged as Scottish Labour’s justice spokesman, Daniel Johnson, quizzed ministers about dangerous dogs legislation.
He requested information on the number of dog control notices issued under the Control of Dogs (Scotland) Act 2010 and the number of people charged and convicted under the Act.
Legal affairs minister Annabelle Ewing said the number of notices had grown year on year from 92 in 2011/12 to 290 in 2015/16, the last year in which figures are available.
In total, there were 1,059 dog control notices issued over the five years.
Over the same period, Ms Ewing said 102 people were charged with at least one offence under the Act and there were 23 convictions.
There were a further 35 people charged and nine convictions in 2016/17.
Meanwhile, only two people have been disqualified from keeping dogs and there have been no instances where a dog was ordered to be destroyed.
Mr Johnson said: “I believe these figures suggest that there ought to be an investigation into whether the law around the control of dogs is being applied fully or whether it needs to be updated.
“With just two people disqualified from owning or keeping dogs as a result of the 2010 Act, it is worth examining if this law is really working.
“I have been contacted by a variety of groups who are disappointed in the way the law is working: from posties at the CWU union to farmers worried about livestock to the SSPCA.
“Specifically I am concerned by the so-called ‘one free bite’ rule which sees owners given the benefit of the doubt on first offences. In reality we know there are tell-tale behavioural signs, and responsible owners must ensure dogs do not cause harm to others.”
The CWU said some 2,500 members north of the Border have been attacked by a dog over the eight years since the Act came into force.
National health and safety officer Dave Joyce said they would relaunch their Bite-Back campaign, below, to have the laws revised and added: “The situation in Scotland is farcical.
“We have set out to the Scottish Government what needs to be done to start addressing the issue and we will be meeting ministers to discuss it.
“One big problem in Scotland is the total misapplication of the law and the enforcers who are following a one free bite rule.
“You don’t let a murderer have one free homicide before prosecuting and you don’t let a dangerous driver off scot-free the first time he kills a pedestrian. It’s ludicrous to apply such a rule with dog attacks and it gives irresponsible dog owners the green light to adopt a couldn’t care less attitude.”
He added there was no time for the SNP government to dawdle on dangerous dogs and continued: “There’s been an 80 per cent increase in dog attack hospitalisations in Scotland.
“Postal workers, children and other workers are all being attacked and injured and the situation is frankly out of control. The problem we face is the failure of the law to deal with the problem and failure of the police, procurator fiscal and courts to deal with irresponsible dog owners effectively and give it the priority it deserves.
“In the years since the Act came into force only two people have been disqualified. It’s preposterous.”
Last month, former health secretary Alex Neil MSP said 1,057 adults and children received hospital treatment in the NHS Greater Glasgow and Clyde area for dog bites in the first six months of last year and called for a “post-legislative review” of the Act.