Through proper and careful herbicide treatments, hand pulling, mulch, and persistence you should be able to get control of the weeds.

With grassy weeds, assuming they are true grass species and not sedges, you can use products that have sethoxydim or fusilade as the active ingredient. These chemicals are specific to grasses and should not harm the periwinkle and will not control broadleaf weeds. 

For the broadleaf weeds, hand pulling may be your best option for those growing amid the periwinkle. There are no selective herbicides that you can use as over-the-top sprays that will target just the broadleaf weed plants and not harm the periwinkle. 

Products that have glyphosate as the active ingredient will kill both grassy and broadleaf weeds but could harm your periwinkle if you are not careful. Spot spraying areas where no periwinkle is growing should work well. You can also use a wick or hand application, according to herbicide label, as a direct application to the leaves of the weeds, again being careful to not get the herbicide on the periwinkle leaves. Be sure to read and follow all label directions if you choose to use an herbicide.

Mulch can also be very helpful in reducing and controlling weeds.