The short answer is it depends. It depends on the invasive species; how quickly it reproduces, how many offspring it produces, how many ways it has to spread itself, and what tools it has to overwhelm native species. It also depends the environment it is invading; how large is the ecosystem being invaded, how well suited it is to the invading species, as well as the relative strength of native species defenses. Above all, it depends on how soon the invader is identified and effective counter measures are taken to control the invader.
Invasive species often go undetected for long periods, sometimes decades, until they form monocultures that totally transform a landscape. When this happens eradication becomes impossible and control becomes the only viable strategy.
Even really tough invasive species can be controlled and even eradicated if they detected early and effective steps are quickly taken.