A simple but highly effective way to get rid of under eyelid bags is with filler below the bags.  The bags that begin to show as we age have really been there all along but as we lose volume in our face. the bags, which are pockets of fat around our eyeballs become more apparent. While many surgeons like to operate and take this fat out, this is, in fact, contributing to the aging process in the long run.  Simply replacing the volume below these bags can produce a pleasing result without the complications of blepharoplasty.
My preferred filler for this area is Bellafill.  This is a long term filler that produces marvelous results.  Here are many examples of before and after photos using Bellafill to correct under eye bags.