Weller, 55, sued Associated Newspapers for misuse of private information on behalf of daughter Dylan, who was 16 when the seven unpixellated pictures appeared on MailOnline in October 2012, and twin sons John-Paul and Bowie, who were 10 months old.
The one-time frontman of The Jam and the Style Council was not at London's High Court to hear the ruling by Mr Justice Dingemans.
The pictures were published after a paparazzo followed Weller and the children on a shopping trip through the streets of Santa Monica, California, taking photos without their consent despite being asked to stop.
Associated Newspapers argued that they were entirely innocuous and inoffensive images taken in public places and that the Wellers had previously chosen to open up their private family life to public gaze to a significant degree.
Weller gave evidence that he did not volunteer information about his family when he spoke to the press to promote his records but was a candid person who would answer a question if asked.
He said: "My preference would be just to talk about my music but I can also see that would be a very dull interview. It's just chit-chat. There's a big difference between that and someone following you around and taking photos of babies. That's a distinction that needs to be made."
The singer said he was never happy about his children being in the papers but he could not make a court case of it every time it happened.
"They overstepped the line with the photos in LA, where they are full frontal pictures of the babies... I don't think the children should be brought into it, not until they are old enough to make their own decisions."
He said it was incorrect that Dylan, who was in one shoot for Teen Vogue when she was 14, was a model and she had been "entirely intimidated" by the paparazzo who took the photos without consent.
"Even when I asked him to leave, and I thought he had left, I came out and he is still taking photos of a very frightened 16-year-old holding her baby brother. What kind of person is that anyway?"
He said he was relaxed about his 27-year-old wife Hannah putting pictures of the twins on her Twitter account, which had 3,570 followers, as long as their faces were not shown.
Mrs Weller told the court: "The image of their face should be controlled by their parents and not on a national website. It is part of my job as a mother to control who sees that information."
Ruling that there was a misuse of private information and a breach of the Data Protection Act, the judge awarded £5,000 to Dylan and £2,500 each to John Paul and Bowie.
He said the Wellers, whose lawyers had asked for at least £15,000 for each child, would have refused to give consent for the photos showing the faces of the children if asked and this was their consistent approach to dealings with the media.
The judge said: "I accept that Paul Weller and Hannah Weller had general concerns about security in relation to their children, but had no specific evidence for those concerns.
"They had the normal concerns of any parent for the safety and security of their children, heightened by the fact that Paul Weller was well known. I accept that the main purpose of Paul and Hannah Weller in bringing the action was to attempt to ensure that the children were left alone as they grew up.
"In my judgment, the photographs were published in circumstances where Dylan, Bowie and John Paul had a reasonable expectation of privacy. This was because the photographs showed their faces, one of the chief attributes of their respective personalities, as they were on a family trip out with their father going shopping and to a cafe and they were identified by surname.
"The photographs were different in nature from crowd shots of the street showing unknown children. The photographs showed how Dylan, Bowie and John Paul looked, as children of Paul Weller. The photographs also showed how Dylan, Bowie and John Paul looked on a family day out with their father."
The balance came down in favour of finding that the right to respect for private and family life overrode the right to freedom of expression.
"These were photographs showing the expressions on faces of children, on a family afternoon out with their father.
"Publishing photographs of the children's faces, and the range of emotions that were displayed, and identifying them by surname, was an important engagement of their Article 8 rights, even though such a publication would have been lawful in California.
"There was no relevant debate of public interest to which the publication of the photographs contributed. The balance of the general interest of having a vigorous and flourishing newspaper industry does not outweigh the interests of the children in this case.
"I consider that, although the interpretation of the Editors' Code is not for me, this conclusion is consistent with the approach set out in the Editors' Code, which recognises that private activities can take place in public, and that editors should not use a parent's position as sole justification for the publication of details of a child's private life."