The apology comes after John Major, a retired judge, issued a scathing 4,000-page report about Canada's handling of the attack.
Harper called the report's findings "deeply disturbing".
Major's report said a "cascading series of errors contributed to the failure of our police and security forces to prevent this atrocity" and made 64 recommendations, including reforms to Canada's security services.
Perviz Madon, who lost her husband in the attack, has said she hopes that Major's recommendations become law.
"We have gone through the ups and the downs in these past 25 years and the recent release of the inquiry report has brought about some semblance of closure to most of us," she said.
"We are hopeful that these findings will bring about legislative changes so that something like this never happens again here in Canada."
Like most of the attacks' victims, Madon is a Canadian of Indian ancestry.
'Conceived in Canada'
In his apology, Harper touched on an underlying complaint from the families of victims: that the attack was seen as an Indian tragedy, not a Canadian one.
"This atrocity was conceived in Canada, executed in Canada, by Canadian citizens, and its victims were themselves mostly citizens of Canada," he said.
"We wish this realisation had gained common acceptance earlier."
|Of those aboard the Air India jet that exploded off the Irish coast, 82 were children [AFP]
The aircraft bombing was allegedly carried out by Sikh fighters who were aiming to gain an independent homeland in India.
To this day, only one man - Inderit Singh Reyat, an electrician from Duncan, British Columbia - has been convicted in the case.
He pleaded guilty to a lesser charge in relation to a second bombing attempt at Narita airport in Toyko, Japan, which killed two baggage handlers.
Ripudaman Singh Malik and Ajaib Singh Bagri, two Canadian Sikhs, were charged in 2000 with the murder of those on board, but in 2005 a provincial supreme court judge found that the evidence against them was not credible.
After the attack, Brian Mulroney, then Canada's prime minister, called Rajiv Gandhi, his Indian counterpart, to offer condolences on the deaths.
But Mulroney did not call the victims' families in Canada to do the same.
The apology was "nice to hear", Mandeep Cheema said. Her father was killed in the attack.
"At the same time, it does not change anything," she said.
"They keep saying, not again. I suppose the apology is good if it means that".