Depending on how many purslane plants in the lawn (and how large of a lawn) you're actually talking about... if it's too many plants to go around and individually dig up, you may have to completely re-do the lawn, in my opinion (making sure the purslane is under control before you re-plant). I think if you just mow it off it's going to spread even more; and you definitely don't want to let it go to seed. You could also try contacting a lawn service and see if they have a solution to offer.
