Richard Cassidy stabbed David Farish 14 times and slashed his throat at his home in Tweedbank.
A court heard Cassidy had been told the 75-year-old had abused a close friend when she was a child.
He had earlier offered a guilty plea to the reduced charge of culpable homicide but was convicted of murder after a trial at the High Court in Glasgow.
A jury heard how the ex-soldier flipped when Mr Farish told him the police "could not prove anything" after he confronted him in February last year.
Cassidy went on to hand himself in - but said he did not want to be known as a "cold blooded killer".
He now faces a life sentence when he returns to the dock next month.
The woman who accused Mr Farish of physically and sexually abusing her gave evidence during the trial.
She had been friends with Cassidy for a number of years and she confided in him.
She claimed she was regularly preyed upon starting when she was seven.
As well as the sexual accusations, it was alleged Mr Farish once broke his hand having hit her so hard.
The court heard police investigated in 2006, but no charges were brought.
The woman recalled how Cassidy had been "understanding" about how she had apparently suffered.
It was last February when he decided to confront Mr Farish at his home where he was alleged to have told him police could not prove anything.
Cassidy went on to murder him, leaving him to bleed to death.
The woman who accused Mr Farish later learned he had died - but did not initially think Cassidy was the culprit.
She told the jury: "All I could think of was that he had met his match this time."
However, she recalled being "gobsmacked" when Cassidy confessed he was responsible.
The woman told prosecutor Angela Gray: "He was really sad. I think because I have never seen a side like that in him.
"I could not believe what he was saying... that he was able to take a life. It was not his nature."
She later drove Cassidy to the local police station where he handed himself in.
He went on to send a string of letters to the woman from prison while on remand.
In one, he said there must have been a lot of "hatred and anger" in him to do what he did.
He added it was Mr Farish's apparently "cocky" comment which made him lose control.
The woman said she did not intend to speak to Cassidy in the future.
Judge Lord Summers deferred sentencing for reports.
After the verdict, Det Insp David Pinkney said: "This was a particularly heinous murder of a frail, elderly man in his own home.
"Thanks to a thorough and robust investigation by the major investigation team we were able to quickly identify Richard Cassidy as the suspect.
"Information we received from the community was also crucial in bringing Cassidy to justice.
"This conviction will not reverse his cruel actions but I do hope it will bring some measure of resolution for Mr Farish's family."
A 70-year-old man has been found guilty of murdering a fellow pensioner in the Borders.