Since Chickweed can root from any bits of stem and leaf left behind you need to be sure you get it all when pulling it out or it will simply grow back. Over the years I have had good success controlling Chickweed and Purslane by covering them with newspaper and covering the newspaper with a mulch which deprives those plants of access to sunlight and they die when they have none, putting the nutrients they removed from your soil back into the soil.

Applying Preen to the soil before it germinates and starts to grow will give good results for the season (Don't plant any seeds in the soil for 6-12 months or they may not grow, either). Preen can work well if the weeds are growing among perennials or transplants.

Once the chickweed is growing, 2,4-D or Round-up will take out the current growth, then mulch heavily to to keep new germination from getting light. If it is growing with broadleaf perennials or transplanted flowers, the 2,4-D or Round-up could take them out, too. In that case, stick to mulches or hand weeding or tillage.

Both of those "weed" killers are gross overkill for most "weeds", and are not really needed. Both unless used with great care in the garden can kil loff any desireable plants as well since they are both broad leafed "weed" killers.