Earlier this month, Salop began the search for a new boss following the departure of Micky Mellon, and they turned their attention to Hurst, who has been in charge of Grimsby since 2013.
The League Two club announced on Monday that Shrewsbury made an official approach for Hurst at the weekend and after talks, compensation has been agreed.
A statement from Grimsby read: "We wish to place on record our thanks to Paul for all he has done for the club since his appointment in March 2011. He will forever be remembered as the manager who successfully saw the return of this club to the Football League. He leaves the club in a strong position and we wish him well for the future.
"Assistant manager Chris Doig will be joining Paul at Shrewsbury Town and again we thank Chris for his work for the club. Our attention now turns to the appointment of a successor, with work already commencing."
Dave Moore will take charge of the first team on an interim basis.