It grows very rapidly from fresh seed, and that sounds like the way to go for you. This plant is a tender perennial, by the way, not an annual plant, and under the right conditions can live for several years. However, most of us grow it AS an annual, either buying new plants or a packet of seed.

I am not sure how they grow from cuttings, though I expect that if you search around in your plant, you'll find some stems with rootlets already emerging from the nodes. That would be the place to make your cuts and plant them in a good potting mix. I never suggest that cuttings be started in water.

They have rooted themselves from where they've touched the soil in my greenhouses, although I usually grow them from seed. I have taken cuttings and they do well, but one must watch out for stem rot, and botrytis in the foliage if allowed to mat up and stay damp.

This propagation method works for warmer climate areas where you do not get a winter frost. At the end of the season (around October), the plant growth gets to look dried out and not attractive. At that point, you can just clip the plant at the edge of your container. The remaining plant stems in the pot will send out new growth in the spring. Just place the plant in a part shade or bright light location and water it when it looks dry. The plant will look like it is just dead stems, but it is still alive and it will grow again in the spring.

Propagation from the mother plant:
This plant is often used at the edge of container plants and gives a nice water fall effect down the side of the pot. When grown in containers that are on a plant stand, the plant can reach the ground and continue to grow. At this point, you can place the "excess growth" that is on the ground in a 4" plastic starter pot that has potting soil in it. This plant has tendrils (tiny roots at the base of leaves that are on the stem). These will take root in the starter pot. Here is the method for doing this type of propagation:
1. You will use cut paper clips to hold the stems of the plant down in the pot. Each pot needs about 10 paper clips. Open the paper clip so it is in an "S" shape. Use a pruning shears to cut the paper clip in half - so you have 2 "U" shaped pieces. Put the metal pieces in a small container so you have them ready for use.
2. Use fresh potting soil. Fill (almost to the top) one or more 4" plastic pots. Green or black plastic pots blend in better among your other flowers.
3. DO NOT CUT the EXTRA GROWTH FROM THE PLANT. Place the current extra growth across the top of the 4" pot so you can see how many stems will be used for each pot. It is better when you have at least 6 extra inches of plant growth that was on the ground. Each pot will use 8 to 10 plant stems. Take the 8 to 10 stems off the top of the pot. You were just "sizing up" how many stems to place in the pot.
4. Working with one stem at a time, place the stem across the top of the soil. Locate 1 or 2 tendrils on the stem. The tendrils are at the base of the leaves. They may be very small. Use the U-shaped clips to secure the stem in the soil at the location of the tendrils. Also use the clips to secure the stems in the soil near the edge of the pot. You will use 3 or 4 clips per stem. Use this method to secure the 8 to 10 stems in the soil.
5. Next place a small amount of potting soil at the location where you put in the paper clips. This soil should be moist. If you happen to have some worm castings, add some of that to your moistened soil. This works as a fertilizer, but is not necessary.
6. Use a small watering can, and give the pot a small amount of water. Place the 4" pot at the base of the mother plant. Water the new starter pot when you see it is dry. This plant likes slightly moist soil.
7. Use the same method with additional pots until you have potted all the extra growth that was on the ground.
8. It takes about 6 weeks for the tendrils to grow to a strong root base in the starter pot. At that point you cut the plant stems from the mother plant and you have a new independent plant.
Additional comments:
1. This works because the starter plant is actually growing from the original mother plant and no cutting has taken place. I had no luck whatsoever with cuttings.
2. Using dark green or black 4" pots makes the starter plants "sorta" blend in with your other plants. Most people don't notice that the starter plant is still attached to the mother plant.
3. This plant looks good for the complete growing season and does well in bright shade and part sun. I don't have full sun but I think it will do well there also. I like using lobelia in containers, but lobelia often dries out or dies in temperatures over 85 to 90 degrees or in full sun.
4. When the new plant is started from the extra growth of the mother plant in the middle or end of the growing season, the new plant will look good throughout the winter in warmer plant zones (zones 7, 8 and 9).
5. I often use Silver Falls Dichondra with Dusty Miller, geraniums and Crystal Palace Lobelia in a container. It also looks good with begonias, impatients, and cyclamin.
6. This plant is often used as a trailer at the edge of a container. When you have your new plant in a 4" pot, you can remove the plant from the pot, pry it in half and have a narrower plant to transplant at the edge of a container. When clips were used to secure the stem at 2 locations (near the edge) of the 4" pot, tendrils take root at 2 locations. Thus, you usually are not disturbing the root base for the plant when you pry it apart to get a narrower plant.
7. Two years ago I started with three 4" plastic container plants that I purchased from the greenhouse for $4.50 ea. I now have 12 containers with Silver Falls used as a trailer. I also gave 3 4" pots to a neighbor.
8. Time: It takes 4 minutes (or less) to secure the extra growth in a 4" pot.
9. Cost: The cost of some paper clips, potting soil and 4" plastic pots (which you likely already have).
10. Success rate: 100%
11. What doesn't work: clippings in water do not work. Just laying the stems across a pot and covering with a little soil does not work very well - you have to secure the stems with paper clips at the location of the tendrils and at the two edges of the pot.

Discovered this in 2016 and you've probably long-forgotten -- but thank you SO MUCH for this detailed information; so helpful!

Still reading in 2019! I have 8 silver falls I’m over wondering and concur with the propagation strategy above. You also don’t need the U shaped holders, just put a pinch of dirt over a portion of the stem that doesn’t have leaves and that’s enough. SF loved to root.

I have a large LED light above my SF and water one very other week or so. Am struggling with root rot, nats, and the stems of my 5ft long vines getting dry and bending. I have A LOT of new growth at the base of the soil, likely from the light. I will start spraying the plant with water and doing moderate light to balance keeping the long vines and new growth. I know these plants will live until I get them back into window baskets after winter, I’m just not sure how long the surviving vines will be. Either way, treat them as a normal house plant over the winter (make sure they get some light- if the leaves lose their color, they need more light- buy a $10 light on amazon)and they will survive. I did not trim my vines in hopes they would grow even longer. The longest is probably 6-7 feet.