Pistorius, 26, known as the "Blade Runner", was charged on Friday with murdering model and law graduate Reeva Steenkamp in the early hours of the previous day.
He broke down during a 40-minute bail hearing at a Pretoria court, but was not asked to enter a plea.
"They had plans together and Oscar was happier in his private life than he had been for a long time," Anthony Pistorius said in a statement released by his nephew's agent. "We are in a state of total shock – firstly about the tragic death of Reeva, who we had all got to know well and care for deeply over the last few months. All of us saw at first hand how close she had become to Oscar during that time and how happy they were."
The suggestion runs counter to comments made by the athlete's father, Henke, who was quoted in the New York Times as saying he had never met his son's partner: "I don't discuss my son's relationships. I have, in fact, not met the lady."
Prosecutors alleged the shooting was premeditated, a charge that could put Pistorius behind bars for life if he is convicted.
Police have said there are no other suspects and the pair were the only people in the house at the time.
Initial reports suggested he may have mistaken her for an intruder.
Anthony Pistorius reiterated the family's belief that the track star – a double amputee who became one of the biggest names in world athletics when he ran at last year's London Olympics – had not shot Steenkamp deliberately.
He said: "We have no doubt there is no substance to the allegation and that the state's own case - strongly refutes any possibility of a premeditated murder, or indeed any murder at all."
Pistorius is being held in a Pretoria police station before the resumption of his bail hearing on Tuesday. He is "numb" with shock and grief, the statement said.
Several South African media reports have said Steenkamp was shot through the bathroom door and was hit four times.
Police said on Thursday that witnesses had heard disturbances at the home before the shots, adding there had been previous incidents of a "domestic nature" at the home.
Although the public image was of an easy-going young man, a Twitter posting by Pistorius in November paints the picture of a would-be action man obsessed with security.
"Nothing like getting home to hear the washing machine on and thinking its an intruder to go into full combat recon mode into the pantry! waa," read the tweet.
Meanwhile, South African state broadcaster SABC said it will air the first episode of a reality show featuring Steenkamp that was filmed last year in the Caribbean.
Her cousin, Sharon Steenkamp, said the family had not been consulted but they were not against it being aired because she "was proud of being in the show".
Steenkamp's uncle, Mike Steenkamp, said there would be a memorial service for Reeva in her home town of Port Elizabeth.