Moore, who has spent four-and-a-half-years in Gloucestershire, was offered the role at Town almost a week ago and has since spent time deliberating on whether to come to Shropshire.
But he has made his decision and has accepted the role at Montgomery Waters Meadow.
During his time at Cheltenham, Moore took the club from a relegation-threatened League Two side to achieving two of their best finishes in the EFL.
They also won their first EFL title when they became League Two champions at the end of the 2020/21 season.
The appointment is the start of Shrewsbury’s summer business as they build for next season with decisions needing to be made over the coming days.
Steve Cotterill’s future still remains uncertain after the club announced they were creating a new management structure without consulting the Town boss.
Speaking after Salop’s 1-0 loss to Lincoln on the final day of the League One season – the boss revealed he was not sure what the future holds for him – despite the club’s second-highest finish in 33 years.
A situation that 10 days later has still not been resolved.
Cotterill is not the only one whose future is in the air – as Town players who are out of contract are still waiting for the club to make a decision on whether they have a contract for next season.
In a statement released by the club, Cheltenham chairman David Bloxham paid tribute to Moore's work in Gloucestershire.
He said: "Micky leaves us after nearly five successful years and we can only thank him for his excellent contribution during that time.
"He has played a key role in the football side of the operation and helped us to develop from a club looking over its shoulder in League Two to one capable of holding its own against the likes of Sunderland and Sheffield Wednesday. I would like to wish Micky all the very best for his new role and I hope that we will continue to maintain a good relationship with him for many years to come. He will always be welcomed back at Cheltenham Town.
"It's no secret that we wanted to keep Micky with us but he felt that the time was right for a new challenge."