Then dump mulch and compost on top of the newspaper. The grass will die underneath and provide food for worms and other friendlies. This is what I did with my forsythia, except I didn't bother with the newspaper because I didn't care if the grass grew up around the stems of the plants. Now, no-till is not perfect. Uh, no, the bad news is it may not work well for root crops the first year.
Carrots can divide if they hit large pieces of stuff like shredded leaves.
Other root crops can hit the newspapers and be forced to grow sideways. You will probably have to remove the top layer of sod with the noxious weed and start with the bare ground. Of course, if these noxious weeds are in your yard, they will invade your garden, and you'll have a never-ending battle. But you'll have that battle whether or not you're wreaking carnage on the soil life. So, go out there, annoy your significant other and triple the size of your garden.