Shrewsbury, who have been without a manager since Micky Mellon left on October 7 to take over at National League side Tranmere, approached Grimsby over the weekend for permission to speak to Hurst and the Mariners confirmed his departure to the New Meadow on Monday morning.
Chris Doig has also left Grimsby and he will be Hurst's assistant at Shrewsbury, who are currently bottom of League One and six points adrift of safety.
Meanwhile, Dave Moore has been installed as caretaker manager while Grimsby look for a successor to Hurst.
A statement on Grimsby's official website said: "Following an official approach by Shrewsbury Town over the weekend and the agreement as to compensation which would be payable, Grimsby Town gave permission for Shrewsbury to speak to Paul Hurst with regard to the vacant manager's position at Shrewsbury.
"Agreement has been reached between Shrewsbury and Paul Hurst who has been appointed manager.
"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 and again we thank Chris for his work for the club.
"Our attention now turns to the appointment of a successor and work has already started on this
In the interim Dave Moore has agreed to act as caretaker manager and we are grateful to Dave for once again stepping into this role."
Hurst, 42, guided the Mariners to promotion to League Two last season
The club are currently eighth in the table and only outside the play-off places on goal difference
Shrewsbury announced a press conference for Monday at 2pm to officially unveil Hurst as manager
Source : PA