The 41-year-old 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 the fourth tier 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."
Edwards, who also played for Carlisle, Bristol City and Blackpool, said: "I'm ready to put into practice the skills I've learned from 25 years in the game that has seen me work with, and learn from, some superb managers and excellent players
"I believe I can develop a team at Tranmere Rovers the whole club and the whole of Wirral can be proud of.
"My aim is to make sure we are challenging at the top end of the table, as well as setting up a structure to produce more home-grown players.
"I'm looking to move the club forward with an attacking brand of high-energy football and I can't wait to get started."
Tranmere chief executive Jeremy Butler said Edwards filled all the criteria they were looking for in a new manager.
" Our aim was to find a young, hungry manager with the drive to lead Tranmere Rovers forward," Butler said
"We want a manager capable of winning promotion but also one able to improve our current bright young talents, as well as putting in place a pathway to bring through more players from our academy."
Rovers confirmed that M cMahon will continue in his role as assistant manager
Source : PA