Invasive Purple Loosestrife used to choke our waterways. An insect that ate only Purple Loosestrife was introduced, and the stuff is now much less of a problem. There's an example of very successful control of an invasive.
Invasive Multiflora Rose used to choke the hillsides in great thorny tangles. Then a virus showed up (it was not introduced deliberately), and the stuff is way less common now. However, big downside to this one, since the virus also targets garden roses. So, mixed record there.
Invasive Emerald Ash Borer. It's killing native ash trees by the millions. Biocontrols are being experimented with. I hope that turns out well.