Joint managers Anthony Jackson and Bernard Morley left the club by mutual consent last week reportedly after pay disputes, and Alexander has been announced as the duo's successor.
Salford is owned by Ryan Giggs, Gary Neville, Nicky Butt, Paul Scholes and Phil Neville. Co-owner Gary Neville told 'Sky Sports': ''After what happened last week we had to react quickly.''
''Sporting director, Chris Casper, led the process and when he made the club aware of the opportunity to secure someone of Graham's experience and credibility we move decisively.''
''We have acquired the highest quality of manager in Graham to continue the great work that has been done over the last four seasons.''
''We had have stability at this club and we now look forward to a new chapter with Graham, who has accepted a four-year deal at SCFC.''
For his part Alexander said: ''During my first meeting with Chris and then later with Gary, I was immediately impressed by the scale of the ambition this club and the owners have.''
''It has made rapid progress over the last four years and I am delighted they have asked myself and Chris to spearhead the next stage of the club's development.''
''There is a great deal of work to be done on and off the pitch to fulfil the potential of Salford City and I can't wait to get started. Even though this season has just ended, our work in preparation for the club's first season at National level will continue throughout the summer.''