Arshad Mohammed, 48, raped the 19-year-old woman in his Skoda taxi after driving her to a dead-end in Neilston, East Renfrewshire, in November 2010.
He was caught after police took a DNA sample when he was detained in March and April 2011 for making sexual remarks to two 17-year-old women.
Sentence on Mohammed was deferred and he was remanded in custody.
Pakistan-born Mohammed, who holds Italian citizenship, was convicted by majority verdict following a six-day trial at the High Court in Stirling.
The court heard that the rape victim had got into his 50 50 Cabs Skoda taxi in her pyjamas, with a coat on top, following an evening watching television at a friend's house.
Mohammed abducted her and drove the victim to a secluded spot in Renfrewshire where he raped her.
After the attack he told her: "See you after, princess."
The court was told the woman went round to a friend's house in tears and the police were called.
Mohammed, who was described as having "more than one teenage daughter" and a wife in Pakistan, was caught after he targeted two other lone, teenage passengers.
He was detained by police twice, in March and April 2011, after he subjected them both to explicit sexual remarks.
On these occasions he was required to give a DNA sample which was matched to DNA on a swab taken from the rape victim.
Following his arrest, Mohammed appeared at Paisley Sheriff Court in June 2011 and was granted bail but he broke the conditions by failing to report to police, failing to hand in his Italian passport, and failing to appear at the High Court in Glasgow in April 2012.
Mohammed was arrested in Norway on a European Arrest Warrant two years later, and extradited back to the UK in August 2014.
During his trial he had lodged a special defence of consent, and claimed that his victim had asked him if he wanted to have sex.
He claimed she said afterwards, "Thank you driver, now I can get a good night's sleep."
In court, he claimed a "disability" in his legs made it only possible for him to have consensual sex.
The jury also found him guilty, unanimously, of two charges of breach of the peace for uttering the remarks to the 17-year-old girls.
Judge Lord Matthews said there was a "substantial sexual element" to the breaches of the peace.
He deferred sentence for reports and remanded Mohammed in custody.
A Glasgow taxi driver who fled abroad to avoid a rape charge has been found guilty of the sex attack.