Vetter said that one woman, who he had advised to have a Caesarean section a week before hand, had delayed in order to be eligible for the aid.
"There was a small risk but everything worked out okay. It was a healthy boy." he said.
In the build up to January 1, the German media offered many tips on how women could hold off giving birth - from avoiding physical and sexual activity to taking magnesium and homeopathic medicines.
But medical experts urged pregnant women against medical intervention to delay births, saying there was a possible risk of complications and that it would likely have only a minimal impact on the timing.
The German government has for a while been concerned about the country's shrinking population and the new benefits programme offers an incentive, mostly to working parents.
A parent who takes time off from work to care for a newborn can receive two-thirds of their net monthly salary, up to a maximum of €1,800, for 12 months.
"For some, there's a lot of money at stake"
If the other parent takes a further two months off, the benefit is extended to 14 months.
Germany has a population of 82 million but a low birth rate has meant the average age of the population has increased.
The low birth rate and a steadily aging population is a problem shared by many highly-industrialised countries and seen as one of the factors hampering the development of what is Europe's largest economy.
Gabriele Meyer, a career mother from Munich pregnant with twins, was due to deliver on January 22 but had a Caesarean section on December 29 at her doctor's orders because one of the twin foetuses was having problems.
"It means more than €20,000 [was lost] to me," she said. Meyer would have got €1,800 per month for a child plus €300 for the second child.
Her husband Jens Meyer, a manager at BMW, plans to appeal directly to Ursula von der Leyen, the German family minister, citing a medical emergency.
But the government has ruled out making any exceptions and rejected direct appeals for leniency from the Protestant church.