Morning glories taking over and out competing garden every year.

I have a larger garden bed where I usually grow sunflowers and perennials. On one end I had grown some morning glories one year and they looked absolutely lush and lovely and stayed contained for about 2 years after seeding down. Before I knew it they had completely taken over the bed chocking out whatever manages to shoot up ahead of them. I manage to stay on top of them for a few weeks in the spring but the number of them is too overwhelming and I would have to spend hours a day pulling them up. Is there any way I can control them without an application of weed killers? I'm looking for something that may out compete them but I wouldn't have to worry about that seeding down or growing again the next year...