Murray, who has served as both a player and assistant manager since 2012, became caretaker last month following the departure of Paul Cox by mutual consent.
In his spell as caretaker Murray led Mansfield to a 1-0 win over Plymouth Argyle and a victory by the same scoreline in an FA Cup first-round replay with Concord Rangers.
Mansfield were beaten 3-0 by Luton Town last Saturday but that has not stopped the club from making Murray the youngest manager in the Football League.
Murray told Mansfield's official website: "After much thought and consideration, I feel the time is right for me to take this step into management.
"I'm excited about this challenge and new adventure and I believe I have the knowledge, ability and contacts throughout the game to be successful.
"This is the proudest day of my career to date."
The 33-year-old's first game as permanent boss comes at Cambridge United in the second round of the FA Cup on Saturday.
