Rodolfo Cardoso took over in the HSV dugout following the sacking of Oenning last month, but under league rules he was not permitted to work for more than a fortnight since he does not hold a suitable coaching licence.
"Hamburg sporting director Frank Arnesen will temporarily take over the duties of the head coach at HSV and responsibility of training the team," read a club statement.
"Arnesen will function in the dual role until a suitable candidate is found for the coaching job."
The 55-year-old Dane said on www.hsv.de: "I will fill the role until the coach is found because that is the best for HSV.
"I went through all the possible alternatives and came to the realisation that the right coach for HSV is currently not available.
"I must also feel that he is suitable, that he can make our team better and that I can work with him.
"I want to have the coach who is best for HSV and not take the second or third choice, just so we have a solution."
Arnesen was tight-lipped on possible contenders for the full-time role.
"I do not want to talk about candidates," he said. "I have a dual role, which I will fill for as long as it needs to be filled.
"Of course this also means a risk for me. I am responsible and I have to be judged. But if there is a benefit for HSV, I'm willing to take that risk."
Cardoso led Hamburg to their first Bundesliga win of the season in his first game at the helm - 2-1 at Stuttgart, before a 2-1 reverse at home to high-flying Schalke.
Arnesen will be assisted by Cardoso and Frank Heinemann, and does not plan to make sweeping changes.
"I will not change much," he said. "We have just produced our two best games this season. We are on the right path."
Arnesen took up the role of Hamburg sporting director in July. He had been in a similar role with Chelsea since 2005 after first arriving in England at Tottenham in 2004.
He previously had a spell as assistant coach of PSV Eindhoven after ending his playing career in 1988.
The Dane played 52 times for his national team and scored 14 goals, and celebrated three Dutch titles with Ajax.