You have two options.

1. The soil is dead, and in time Mother Nature will fix it on her own, but it will be slow.

2. Remove the soil, replace, and reseed.

Last comment is DO NOT TELL anyone about it. If by chance the EPA were to learn of it, you own it and it becomes a very big expensive problem that you own. Does not matter if someone else did the damage, you would be responsible for all the clean up work done using EPA approved contractors and disposal, plus damages for all future claims that you contaminated the ground water. It could easily get out of hand and bankrupt you.

Down here it is routine for people in the country to spray diesel fuel to kill weeds under fence lines. By routine I mean they have to repeat every so often because the weeds grow back. For your issue I'd be inclined to spray molasses and baby shampoo on the areas and cover them with mulch for a few months. I realize that is not a quick fix, but it is another alternative.

The quick fix is as TW described. If you dig it out keep your nose working to be sure you get all the affected soil. But if you get down deeper than a foot, that should be all you need to go. When you put the new soil back in, you should heap it up to allow for settling over the next 3 years. Otherwise you'll end up with a hole.