William McCance, 57, from Baillieston, abused the girl at a house in the Dennistoun area, between 1984 and 1991. He began abusing the girl when she was nine and raped her when she was 15.
McCance was also convicted of sexually abusing her brother when he was aged between seven and eight.
Sentence on McCance was deferred and he was remanded in custody.
The High Court in Glasgow heard that McCance ran a car parts shop in the city's London Road.
The girl never told anyone about the abuse until she decided to speak to someone at the NSPCC and they then contacted the police.
In evidence, her brother, who was abused as a boy, denied making up stories about McCance and said: "I've spent my life trying to forget what he did to me."
McCance claimed that both victims were making up stories about him and denied any sexual abuse.
His defence counsel Dale Hughes said: "He has a family and has been hard-working at his London Road business for many years."
Judge Lord Burns placed McCance on the sex offenders register and deferred sentence for background reports.
A shopkeeper has been convicted of raping and indecently assaulting a schoolgirl in Glasgow.