Krack has been with BMW since 2014. He formerly worked in senior motorsport for Porsche and has a decade of experience in F1 with Sauber, which later became BMW Sauber.
The news comes following the announcement earlier this month that Otmar Szafnauer would vacate the role after 12 years in charge.
Szafnauer oversaw the team, based at Silverstone, through three different guises: Force India, Racing Point and Aston Martin.
Krack will report to Martin Whitmarsh, CEO of the Aston Martin F1 team, leading its "technical and operational functions".
He said: "It's a thrill and an honour to have been appointed to the position of team principal of Aston Martin Cognizant Formula 1 Team, and I'm very grateful to [Aston Martin chairman] Lawrence [Stroll] and Martin for giving me such a fantastic opportunity.
"Aston Martin is one of the greatest automotive brands in the world, and to have been asked to play a leading role in delivering the on-track Formula 1 success that such an illustrious name so richly deserves is a challenge that I plan to embrace with energy and enthusiasm."
Stroll claimed the team is now "on its way" to fulfilling its "lofty ambitions" in year two of its five-year plan.
"Winning in Formula 1 is all about bringing together the best efforts of all the principal players - drivers, engineers, mechanics, everyone - and Mike is perfectly placed to do just that," he said.
"We're hiring brilliant new people every week. Our new factory build is coming along very well. We have a fantastic roster of superb sponsor-partners."
Whitmarsh has suggested that Krack will assist in propelling the team to the world championship that it's "so determined to achieve".
"Having worked in very senior motorsport positions at both BMW and Porsche, backed by an engineering background in Formula 1 with Sauber, Mike possesses a blend of experience and expertise that makes him the ideal choice for us," he said.
"He works well with drivers, having prioritised that aspect of his role at BMW in recent years, and also with engineers, since he speaks their language."
Aston Martin will go into its second F1 season with drivers Sebastian Vettel and Lance Stroll, who finished last year 12th and 13th in the standings.