The 49-year-old former Scotland defender takes over from Stephen Robinson, who stepped down last month.
Alexander has previously managed Fleetwood Town, Scunthorpe United and Salford City and will be joined by his long-term assistant Chris Lucketti, who will work alongside Keith Lasley.
He told the Motherwell website: "I am delighted to have been handed this role at a great club. People will get to understand the hard work I put in, along with my staff and players. I have always brought that to every club, as a player and a manager.
“It’s something different. It is something I am really excited about. I have been a manager now for seven to eight years, and had success with the three clubs I’ve been at, but this is out of my comfort zone. I’m someone who is up for a challenge, and I will get that here."
