In a surprise announcement at the weekend, German Chancellor Angela Merkel revealed that Asmussen, 47, would become junior labour minister in her new coalition government, leaving his position on the ECB's executive board just two years into his eight-year contract.
The move appears to be a step down the career ladder for Asmussen who was a top German finance ministry official before joining the ECB at the start of 2012.
Asmussen is responsible for international and European relations on the ECB executive board.
At a hearing in front of the EU Parliament's economic affairs committee in Brussels on today, Draghi described Asmussen's decision to quit as a "tremendous loss for both the executive board and myself personally. We will miss him a lot."
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And a replacement would have to be found "as soon as possible. We can't really afford to operate without a member" at such a crucial time, Draghi said.
Analysts also believed Asmussen's departure could prove something of a headache for the ECB and Draghi in particular.
Asmussen is not the first top-level German official to quit the ECB.
Former Bundesbank President Axel Weber and the ECB's former chief economist Juergen Stark both left early in protest over the central bank's controversial anti-crisis policies.
"Asmussen appears to be leaving for purely personal reasons. But the departure of such an experienced European policy maker ... Is a strategic loss for the ECB," said Schulz.
Asmussen had defended the ECB's policies in Germany's constitutional court against the sharp criticism from Bundesbank President Jens Weidmann.
"This leaves a gap in the ECB's communication of policy in Germany, as well as a gap of communication of the German debate into the ECB," Schulz argued.
Bundesbank vice president Sabine Lautenschlaeger is seen as the most likely candidate.