Depends on the damage being done and the impact/extent/population of the introduced species.
In Western Australia feral cats and foxes (both introduced species) have been aerial baited with bait that native animals (only a few carnivorous species) are not affected by. This has had great success and allowed Woylie's (a native mouse-like species to be reintroduced to areas where predation had eliminated them previously).
Other pests such as pigs are typically only targeted through hunting. Not much else has been effective. However illegal hunters have been known to release smaller pigs into the bush so that they have game to hunt in the future. Similarly wild goats, horses and camels are hunted this way (in Australia a lot of this is done from helicopters).
Then there are chemical controls of introduced species. In Australia rabbits are a bad problem. Two biological controls have been released into the rabbit populations, Myxomatosis and RCV. Both have had limited long-term impact however because of the rabbits mass breeding. Rabbit proof fences have been also established to keep rabbits out of areas. So that could be another control measure.
Other pests still have no effective control measures, such as the cane toad in Australia. Biological controls, that is natural predators were tried in the past but found the native frogs easier to target. A few native birds have developed techniques to kill them, but too few birds ... and as stated biological controls such as this rarely work as native animals and flora (habitats) don't typically inhibit these pests (otherwise they wouldn't be pests).
In regards to introduced weeds the control is typically removal. This may be via a poison that kills the plant directly, or makes it infertile. Typically correct removal techniques need to be adhered to, that is remove the bulb if needed, remove inflorences first if needed etc., and correct remedial work (such as planting native species, erosion control, etc) are required for effective control of weeds.