You can't keep it alive, because it doesn't have roots, but you might be able to take a cutting from it and start a whole new plant. It will be a clone of your original plant, so will be identical.
Cut the main stem off about 1 or 2 inches above soil level. Pot it up in a smaller pot ... I would use a pot about 4 inches in diameter. You have already learned about the importance of using a container that has drainage holes in the bottom. If you keep a drainage saucer under the container, to protect your furniture, be sure to pour off any water that collects in it. You never want the container to stand in water for more than 20 minutes or so.
Dracaenas don't require much water, so be careful not to water them too often. Dig into the soil with one of your fingers to test for dryness. If it's dry down to an inch or two below the surface, then it's time to water again.
I usually recommend a mix of standard potting mix combined with a handful or two of perlite. In this case, however, you should use vermiculite instead of perlite. Dracaenas are sensitive to fluoride, and perlite can sometimes leach fluoride into the soil. Because dracaenas are sensitive to fluoride, you will want to water with distilled water, rather than tap water.
Does that help you?
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams