Jack Shepherd gave an interview to a Georgian TV channel before being arrested
Jack Shepherd was sentenced to six years in July for the manslaughter of 24-year-old Charlotte Brown.
The 31-year-old had been in the Georgian capital of Tbilisi since March and was absent throughout his trial.
Ms Brown's father Graham Brown said: "I feel very emotional at the fact that my daughter will get some justice."
Speaking to BBC Radio 5 live, he added: "I do think the family will be in a much better position to deal with our loss and heartbreak over the last three years."
Mr Brown, who earlier on Wednesday gave an interview to the Victoria Derbyshire programme urging Shepherd to hand himself in, described the fugitive as "a very crass, reckless man who stuck two fingers up to the judiciary system".
"He's done the right thing and thank goodness he has handed himself in," he added.
Shepherd took Charlotte Brown on a date in December 2015
A spokesman for the Georgian Embassy in London confirmed Shepherd's arrest, which comes after Ms Brown's family met with the Home Secretary Sajid Javid on Tuesday.
Under current diplomatic agreements between Georgia and the UK, Shepherd is eligible for extradition.
Georgian Rustavi TV has shown footage of him before he handed himself in, during an "exclusive interview" in his final minutes as a "free man".
Speaking in English, which was then voiced over and translated into Georgian, Shepherd described it as "a tragic accident".
Jack Shepherd let Charlotte Brown drive his speedboat for a "thrill", the Old Bailey heard
Ms Brown's sister Katie Brown said her family were "relieved" Shepherd had handed himself in but described him as "arrogant".
"To just stroll in with a very smug look on his face and to claim innocence is unbelievable. This is a small amount of justice for my sister."
The Ministry of Internal Affairs of Georgia, the law enforcement agency in the country, previously told the BBC it was working with the Met Police to track Shepherd.
The Met said it was informed by the National Crime Agency that Shepherd was in the custody of police in Georgia.
A Home Office spokesman added: "It is now for the Crown Prosecution Service to decide on whether to make an extradition request to the Georgian authorities, via the Home Office."
Charlotte's father Graham Brown said Shepherd was "a very crass reckless man"
After meeting on the dating website OkCupid, Shepherd took Ms Brown on a date on 8 December 2015.
Shepherd spent £150 on wine and food at a restaurant in The Shard before taking her on a speedboat he claimed he owned.
Ms Brown and Shepherd were thrown from the boat when it hit branches in the water near Wandsworth Bridge at about midnight.
Shepherd was found clinging to the hull and Ms Brown, from Clacton in Essex, was pulled from the water unconscious and unresponsive.
A post-mortem examination found she died from cold water immersion.
At the scene in Tbilisi
I am outside the police station where Jack Shepherd, who is now officially under arrest, is currently being held.
He will be moved to a temporary detention centre and his lawyer says according to Georgian law the detention period in this kind of case can be up to nine months.
But it will be up to a judge to decide how long Shepherd will be in the custody of the Georgian police.
Shepherd gave an interview to a local television station where he maintains his innocence.
He says he does not agree with the court's decision and that he is now ready to co-operate with the investigation.
Shepherd made his first appearance at the Old Bailey on 26 January, when he entered a not guilty plea to a charge of manslaughter by gross negligence.
He was released on unconditional bail by Judge Richard Marks QC, but failed to show up for his trial in July.
After his conviction an international arrest warrant was issued.
Despite being on the run, Shepherd has won the right to appeal against his conviction.
Shepherd's solicitor Richard Egan said: "In the light of today's developments I don't think it would be appropriate to comment further until Mr Shepherd is back in the jurisdiction."