I have a bed that was installed with landscape fabric (to prevent weeds) and rocks for mulch. Can perennials in the bed still spread invasively?

Perennials that are invasive got that reputation because they go where they want and take no prisoners. The landscape fabric may challenge them for a time. But most invasive species do their dirty deed with rhizome type roots. The fabric will keep them warm, trapping moisture, allowing them to run until the root finds a weak point. To avoid this problem, plant perennials with a clumping habit rather than a running one. Avoid those that offer “quick coverage” or those that boast “self-seeds.” That seed will sprout in the crevices between your rocks, and the roots will work through the fabric. Remember that landscape fabric is a woven plastic material that allows air and water to pass through. Weeds need sunlight to germinate. The fabric effectively prevents that. The fabric may retard the individual growth of perennials planted through the holes cut in it, but it will not prevent invasive plants from following their natural habit.