With over 20 years of experience, I have found a combination of methods always increases your chances of successfully removing invasive plants and replacing them with desirable vegetation.  Using one method alone usually does not work in the long run so be prepared to employ multiple tactics to control invasive plants. 

Control Method: Example

Physical Control:  Hand pulling, grubing with tools, digging up, using a weed wrench

Mechanical Control: Chainsaw, brushcutter, mower, forestry grinder

Chemical Control: herbicide

Cultural Control: using clean mulch to cover disturbance, cleaning boots and equipment, early detection-rapid response, adjusting mowing height, maintaining dense native plant cover to outcompete weeds

Biological Control: mile-a-minute weevil 

The only way to successfully control or remove an invasive weed infestation from a landscape is to first learn a lot about the specific target species you want to control- you must have an understanding of the species biology, morphology, physiology in order to increase success.  

First step is to identify your target.  ALWAYS identify the organism your are attempting to remove from the system.  Use plant id guides, professional guidance, extension agents, local experts, books, iNaturalist, dichotomous keys, if there is any question about the identification of plants.  Many native plants look like non-native plants (thanks Pangea!) check out this cool guide we use a lot: Mistaken Identity? Invasive Plants and their Native Look-alikes an Identification Guide for the Mid-Atlantic.  

I always rely on and use research based, current best management practices.  Check multiple sources from .gov, .edu., .org websites, books, etc.  I typically review multiple sources and methods depending on the target species. My current go to sources for starting points:  Blue Ridge PRISM, Penn State Extension, Plant Invaders of Mid-Atlantic Natural Areas, King County Noxious Weed Control Program.  Often a search of the target species on Google Scholar will yield research trials related to available control methods.