How To Get Rid of Violets

We just moved into a new house in December. The previous owners were older and, while the wife was an avid gardener, it seems as though she just couldn't keep up with the work around the house and grounds. Being the end of May and having been rather warm for the past two months, the garden has just completely exploded with new growth. Unfortunately, this includes the rebirth of certain "undesirables" as well...

Which leads me to my big question: How do you get rid of violets?!

I am starting to think that the previous owner planted these invasive little pests as a form of ground cover so she wouldn't have to do as much work to keep the weeds down. There was no mulch down and many beds throughout the property have various types of ground cover (pachysandra, "snow on the mountain," etc). In my book, the pachysandra can stay as it is fairly easy to contain despite its energetic will to spread.  I have begun the process of removing the "snow on the mountain" (aka "Bishop weed") and it is thankfully confined to one isolated bed. 

However, these violets are out of control. They're absolutely everywhere: in the lawn, in paving stones, between rocks dividing beds, choking out good plants. It seems like for every one violet leaf I pull (and why can I never seem to get much root?!), four more grow in its place! 

Everything I have read online basically says, "Good luck," or "Learn to love them - you'll never get rid of them." This can't be right. I refuse to accept it. The best advice I have seen for those of us who actually want to be rid of these pesky perennials is to put RoundUp on a cloth and rub each individual plant with it (to avoid hurting the plants I want to keep) or to just keep pulling them out by hand. Our yard is such a size where keeping up with them via hand pulling is not much of an option. I am about ready to try the RoundUp on a cloth suggestion, but I wanted to put it out there first:

Do you have any tricks, products or tips that will help me get rid of (and stay rid of!) this invasive weed without hurting my other plants??