Assuming that there is no other way, may a person falsely accused of something clear his name by revealing the name of the person who actually did the act of wrongdoing or is this considered Lashon Harah?

Yes you may. But only if: 1. You know for sure first hand who did it. 2. Be a hundred percent sure you are not misinterpreting the facts. 3. Speak to that man first so he should confess. 4. There is some sort of damage to you. Some more conditions apply depending on the situation – for more details look up the book Chafetz Chaim chapter 10.