I've got the perfect solution that I've used myself for same problem but just not garlic.  You're right about those sprays like Fabreeze. Only works for a short time and doesn't work for lots of odors. I've got the best solution for eliminating the odor for good... and safe and totally fragrance free. Go out and get a couple of the FRIDGE-IT activated carbon odor absorbers - little purple cube - not the imitations. costs $2 - $3 dollars each and put one in your bag.  Totally unscented so it actually eliminates odor rather than covering up with some nasty fragrance and totally safe to use.
I've used these for worst smells including spoiled fish in car and for closets, fridge etc. Also have two cats and use these on litter box - they are FANTASTIC> No spills or chemicals to worry about. You can buy these in kitchen gadget stores or just go direct which is what I do - contact them at customerservice@innofresh.com and ask for order form. You get it right away.
Definitely the way to go. try it.
Good luck

well if you use a leather cleaner, then you wouldnt have to worry about the leather of the bag being damaged since leather cleaner is safe on leather you can find leather cleaner at stores that sell leather handbags like coach, louis vuitton etc...