Over time, however, there should be evolutionary pressure on predators to prey more efficiently on invasives, while the invasives will not adapt in response. So long as the predators are inefficient, the evolutionary pressure on invasives should be to outcompete within its own species, by reproducing more and reproducing more quickly. A mutation that adapts to predators is unlikely to spread because it will likely impose a cost on reproduction (spend more time hiding and less time feeding, say) that exceeds the benefits of escaping not-very-efficient predation. The effect is a one-way evolutionary arms race.
Only when the predators become efficient at controlling the invasive population, will the invasive start adapting in response, and at that point the invasives won't be that much of an environmental problem.
So, great. Next question is how long does this all take? We've had invasive plants in the Americas for hundreds of years, and invasive animals for a century or more. I'm no expert on these things, but I do read about them and I've not heard of any invasive gradually receding to become a background species. The few that have disappeared or become minor have done so when we've imported biological controls or have adopted scorched-earth campaigns to get rid of the species.
If on a scale of a century or more, very few or no invasives have been brought to heel by the one-way evolutionary arms race, then the time-scale for this to work on most invasives has to be much longer, maybe many thousands of years. This long-term solution helps us understand how nature handles the rare, natural introduction of a new, invasive species. Considering everything else we're doing to screw up our ecology right now, however, the natural solution isn't something we can rely on as a substitute for our own action to deal with the problem we've created.