The 41-year-old Edwards is the permanent successor to Ronnie Moore, who was suspended by Rovers in February after being charged by the FA for breaching betting rules and then sacked last month after pleading guilty.
When Moore was suspended Tranmere were four points above the drop zone, but under the caretaker care of the 61-year-old's assistant John McMahon they won only four of their final 15 games and were relegated to League Two for the first time since 1988/89.
Edwards is the man tasked with trying to lead Rovers back out of the bottom division, with Tranmere announcing that the former Wales international had rejected a position with a top-flight club to take the job.
A Tranmere statement read: "Tranmere Rovers have appointed former Welsh international Rob Edwards as their new manager.
"Edwards, 41, recently left Exeter City and has turned down a job at a Premier League club to take charge at Prenton Park.
"The former Preston defender was assistant manager at Exeter City where he played a key role as the club won successive promotions from the Conference to League One."
- Sports & Recreation
- Tranmere Rovers
- Ronnie Moore