Levein had been hoping to remain in his post until at least the end of the qualification period for the 2014 World Cup.
But, with the Scots bottom of group A and having slipped 10 places to 56th in the FIFA rankings since Levein's appointment, an SFA panel decided to relieve the manager of his duties.
Regan said: "We are taking this decision with real sadness. Craig has worked hard to bring success, and has been thoroughly professional in his approach to the job.
"He has achieved a great deal with the team and in other areas, such as the performance strategy. However, he would be the first to agree that football is a results-driven business.
"We are bottom of the group with two points from four games and we are eight points behind the leaders.
"The view of the board is we are not bottom-of-the-group material, we are better than that. We need to have a new manager to turn around the campaign and move us forward.
"In addition, Craig has stated his intention to leave after the end of the current World Cup campaign irrespective of what happens.
"For that reason we have relieved Craig of his duties with immediate effect."
The Scots have turned to U21 coach Billy Stark as an interim replacement ahead of next week's friendly in Luxembourg.
Regan said: "Billy Stark will take interim charge for the friendly. There will be a short delay in announcing that squad, which Billy will select, and we have informed the Luxembourg FA of this development.
"After the Luxembourg match, the board will meet to begin the process of identifying and recruiting the next national coach to take us forward. We will give ourselves time, given our fixture schedule, to find the right person for the job.
"The challenge for the squad now is to improve our position in Group A and, subsequently, improve our position in the FIFA rankings.
"We also must begin preparations for the UEFA European Championships in France in 2016."