Gobmeier had only been in the post for 10 months, having replaced long-term Ducati boss Filippo Preziosi as part of a wider reorganisation that came into effect at the start of the current season.
Dall'Igna, 47, was an instrumental figure in Aprilia's World Superbike success, presiding over Max Biaggi's two riders' crowns and the 2009 and '11 manufacturers' titles.
Paolo Ciabatti, who was made Ducati MotoGP project manager as part of the early-2013 reshuffle, will keep his position and report directly to Dall'Igna.
The new appointments will become effective on November 1.
Ducati Motor Holding CEO Claudio Domenicali said the changes would bring fresh impetus and direction to the struggling squad.
"Thanks to this appointment, we can look forward to the next racing season with renewed motivation," he said.
"We are confident that the new Ducati Corse organisation, and a strategy even more focused on technical development, will help us achieve our targets, ensuring that Ducati once again becomes a key player both in MotoGP and Superbike.
"I am certain that [Dall'Igna's] solid experience combined with our technology and R&D and the technical support supplied by the Audi/VW Group, will help us restore Ducati to the level of racing excellence it had in the very recent past."
Gobmeier had pledged that revolution was not necessary in order to return to Ducati to the front of the MotoGP field.
This year however both Andrea Dovizioso and Nicky Hayden have failed to score a single podium, and they sit eighth and ninth in the standings.
They also trail non-factory riders from other manufacturers, with Cal Crutchlow - who replaces Hayden next year - Stefan Bradl and Alvaro Bautista all ahead of them.
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