"I will take on both jobs until we can find the coach who is best for HSV," the Dane told the club's official website.
"I want to have the coach who is best for HSV and not just take the second or third option so that we can present a solution."
Oenning, in charge for only six games, was sacked last month after Hamburg's worst start to the Bundesliga.
Former Denmark international Arnesen has previously worked as sporting director at English clubs Tottenham and Chelsea.
Hamburg, bottom with four points from eight games, had attempted to sign FC Basel coach Thorsten Fink but the former Bayern Munich player has a contract with the Swiss champions until 2013.
"I've been through all the possible alternatives and come to the conclusion that the right coach for HSV is not available at the moment," said Arnesen.
Oenning was initially replaced by his assistant Rodolfo Cardoso but the Argentine does not have a coaching licence and was allowed to remain in charge for only 15 days under Bundesliga rules.
Arnesen, who obtained a coaching licence in the Netherlands, has worked as assistant coach at PSV Eindhoven and had a stint with Wilhelm II Tilburg in 2002.
"Naturally, there is a certain risk for me," he said. "But if it is good for HSV, then I'm prepared to take it."
