A method to incorporate leg elastic members into pant-like disposable absorbent garments, and garments made thereby. In particular embodiments, the method includes supplying elastomeric film laminate first and second body panel webs spaced apart from each other. A continuous first leg elastic member is attached to the first body panel web, and a continuous second leg elastic member is attached to the second body panel web. A continuous leg elastic covering web is attached to both the first body panel web and the second body panel web so as to sandwich the first and second leg elastic members between the leg elastic covering web and the respective body panel web, thereby creating a composite web. In particular embodiments, the present invention can provide an efficient and process friendly technique to cover and hold in place leg elastic members in a pant-style disposable garment that employs elastomeric film laminates.