Which perennials are they Dave? Artemisias, euphorbias??
I always keep an eye on any new plants that are notorious for running but even then still get caught out by thinking I will dig them up each year. However, I now avoid all of them.
Glyphosate is best treatment generally but it does depend on the plant. Many can be traced close to the "donor" plant and therefore dug out emtirely....viz., the roots can be traced and removed with further excavations. Others though do run far and wide sending roots down to appear several feet away. These are best treated with glyphosate. Too late inthe year now but next spring watch out for the new growths and spray. Throughout the summer spray any further growths