Slicing and dicing weeds with a hoe works best when the soil is relatively dry, and the same goes for cultivating with a tiller. With their tops mangled and roots cut, most young weeds will quickly shrivel up and die. Be careful to cultivate only the top inch or two of soil or you may injure nearby garden plant roots and drag new weed seeds to the surface. A sharp hoe works much better than a dull one, so refresh the edge on your hoe with a steel file between weeding sessions.
After using either a hoe or tiller to cultivate weeds, go back the next day to nip out any survivors. When battling perennial weeds, you can weaken the plants by chopping them down with a sharp hoe, but it's best to combine hoeing with digging to achieve good control. Never use a tiller in soil that is infested with bindweed, quackgrass, or other weeds that regrow from small pieces of root; they are easily spread by rototilling.
