The former Leeds and Liverpool forward was appointed by Crawley in May 2017 and led the club to a 14th-placed finish in Sky Bet League Two last season.
Kewell replaces Kevin Nolan after he was sacked with Notts County bottom of the Football League.
The Australian will be joined at Meadow Lane by assistant Warren Feeney, while the Magpies are on the verge of appointing a 'vastly experienced' technical director.
Darren Fletcher is to stand down from his role as vice-chairman.
Owner and chairman Alan Hardy told nottscountyfc.co.uk: "Having now worked in football for a significant period of time, I have had the opportunity to learn from other clubs what good looks like.
"I have therefore decided to appoint a technical director, who will set the club's overall footballing philosophy and be on hand to support and mentor the manager as required.
"Harry will, however, be solely responsible for our first-team squad, having the final say on transfers, tactics and team selection.
"He has enjoyed an excellent start to his managerial career and I am hugely impressed with his knowledge and attention to detail."
Kewell's first game in charge of Notts County will be at Grimsby in the Checkatrade Trophy next Tuesday.
Mark Crossley and academy manager Jon Goodman have been handed the reins for Saturday's League Two fixture at home to Forest Green.
Crawley thanked Kewell and Feeney for their services and wished them well after their season in charge.
Experienced players Jimmy Smith and Filipe Morais have been tasked with preparing the team for the trip to Oldham.
Director of football Selim Gaygusuz told crawleytownfc.com: "Harry Kewell made it clear earlier this week that he wanted the opportunity to speak to Notts County which we granted.
"Matters concerning compensation for the remainder of his and Warren Feeney's contract have now been concluded successfully.
"The search for a new manager is under way and we are looking to make an appointment as soon as possible."