The woman, who called herself 'Jessie' asked for money but was declined by her friend. The woman then used a telephone to try and contact a friend in Mosgiel but could not reach the friend.
Ms Thompson said her brother-in-law then drove the woman to a petrol station while her daughter was left at the home as no car seat was available.
"She was prepared to leave the baby with us, and go off without her. Wasn't in the least embarrassed by being left without money, not unapologetic in any way."
The young woman returned as she had no money to buy petrol, but said she may have enough petrol to drive to the petrol station. She again asked to borrow $10 for petrol and was given it.
Ms Thompson said the woman had made no attempt to pay back the money, and her phone message said the number was no longer available.
Since that incident she had heard of six other Dunedin pensioners who have been targeted by a well-spoken young mum - thought to be in her late teens and early 20s - and who had been given between $10-$20.
She advised people to be "jolly careful".
Senior Sergeant Bruce Ross warned people not to be "so gullible and hand over their hard earned pension".
Anyone who may have been duped in the scam should contact their nearest police station, he said.