Choose the scion, also called budwood, that you want to graft to the rootstock. The diameter of the dracaena scion should be equal to the diameter of the rootstock, or as close to equal as you can get. Cut the scion off the "donor" dracaena with the knife. Wrap the cut edge in wet paper towels.
Cut a diagonal notch into the rootstock's stem where you want the scion to grow. Make sure the cut is deep enough to expose the inner tissue of the stem without cutting all the way through the stem. Leave the flap formed by the cut in place.
Unwrap the scion and use the sharp knife to cut off the bottom of the scion's stem diagonally. Insert the cut end of the scion into the cut on the dracaena rootstock. Try to make the two cut edges fit together as closely as possible. Trim the scion stem, if needed to form a better fit.
Press the flap of bark back into place to help hold the scion cutting. Wrap the graft with grafting tape and seal the grafted area with grafting wax. Both products are available at most gardening centers.
Care for the dracaena as usual, keeping the plant evenly moist but not wet. Stand the pot on a saucer filled with pebbles and water to increase humidity around the plant. Cut back on watering in winter. Place the dracaena in bright, indirect light. Dracaenas tolerate some fluctuation in temperatures, but keep the plants out of drafts. Fertilize lightly with diluted, liquid houseplant food.