Prosecutors said David Lomax forced himself upon the victim whilst visiting her home in Leeds in October 1978.
David Lomax accused of 'blackmailing' woman into sexThe defendant, who worked as a warrant officer with West Yorkshire Police, was accused of putting his hand on the mother's shoulder after hearing she was unable to pay a fine and telling her: "Well, we could solve this."
Sophie Drake, prosecuting, told how Lomax then led the woman into her bedroom and raped her.
During a three-day trial, jurors heard that the pensioner, of Esher Grove in Waterlooville, Hampshire, had been arrested at the time but there was insufficient evidence to charge him.
But, after being re-arrested last year and eventually charged, a jury at Leeds Crown Court convicted Lomax by majority verdicts of rape and misconduct in public office.
The pensioner, wearing a grey suit and aided by a walking stick, looked emotionless as he was told the verdicts.
He will be sentenced on Friday.