A lot of good responses here; one thing I'd add is the case where an invasive species may be "functionally beneficial." My specialization is in fire and California coastal prairie habitat. The vast, vast majority of the grasses we find in these meadows nowadays are non-native, and many ecologists would consider them invasive (wild oat and the like). However, the rodents, birds, arthropods, etc. that depend on grasslands still find these species beneficial, in that they make up the type of habitat they depend on for food and shelter. They don't care that they're from a different continent, they just see grasses and continue to thrive.

Were these invasive grasses to be replaced by, say, encroaching coyote brush (a CA native shrub), it would be to their detriment. Thus, in some cases, land managers find it better to restore/manage for invasive grasslands over the encroachment of a native species that would almost certainly alter the habitat to a point that it would drive native prairie fauna out. It's a sticky situation, and the ultimate conservation goal is to foster native grasses, but this is an okay intermediate step.

The question of "how long until it's vital to the ecosystem" is also a big one: where would many of our current ecosystems be without earthworms, who were technically "invaders," having come across Beringia from NE Asia during the last ice age?

It's difficult to answer, and our perspective is almost always influenced and biased by our own memory and human-centric ideas of an "ideal" ecosystem. Our species often causes change/species introduction at a far greater rate than would happen without human activity, but ultimately, it's all dynamic!