Kavanagh continued her impressive form by finishing fourth in a fast race, missing out on a podium place by just one second.
The Hollywood Athletics runner stopped the clock in 14:51, setting a new South African record and producing one of the best performances of her season.
Kavanagh had told SABC Sport before leaving for Denmark that breaking the national record was one of her main goals.
She says she is proud of the performance after working hard and making sacrifices to reach another personal best.
“I knew it was going to be a fast race today, so my biggest goal was just to go out there and be competitive and see how well I could do,” Kavanagh said.
“Obviously being one second off the podium is a bit bittersweet, but I’m just so grateful for the progress I’ve made over the last couple of years.”
Kavanagh, who also ran a personal best in the 5000m at the Commonwealth Games in Glasgow, says setting a national record while representing South Africa is a special moment.
“I’ve had a really strong season and to put this performance out there today and run a national record is so special. I’m so happy with it and so unbelievably grateful to be here, to represent South Africa and to put that performance together today,” she added.
Kavanagh will remain in Europe for another race but has confirmed that she will not compete in the Spar Women’s 10km Challenge in Gqeberha on Saturday.
That means she will miss the chance to challenge for the R203 000 prize money in the Grand Prix series after winning the opening two races in Cape Town and Durban.
In the men’s 5km, Maxime Chaumeton also produced a strong performance, finishing 10th in 13:13.
Chaumeton was just seven seconds outside his own South African record of 13:06, which he set in Lille earlier this year.