The sentence should have a context in which others fail to finish a task, or make a deal on a business, but a certain people succeeds to make it, and you feel a little surprise about it. In this sentence, it uses "嘴", it usually refers to something related to talking to someone into something. For example, your classmates want to delay the homework but the professor doesn't agree, however, surprisingly, one of your classmates, say Tom, he successfully persuaded the professor and made him change his mind. In this situation, you can say "(这件事)到了他的嘴里，居然成了！” "居然" is used when you feel surprised about something. For example, 你居然会说法语? ——You can speak French? (you feel surprised about it) In addition, if you replace "嘴" with "手", it no longer emphasizes that the thing is done by just talking, it means that the person who succeeds on the thing may have physically done things other than just talking, such as working, studying. For example, you have a very difficult and picky customer. It's very hard to make a deal with him. Many of you coworkers tried and failed, but when Tom showed up and made a deal with him, you can say "到了他的手里，居然成了。"

the grammar or the order of this sentence is not correct. And just only one sentence ,so I don't know its meaning either.Sorry. Or it is to say ‘居然到了他的嘴里’。

There's nothing wrong with this sentence. It means that he is a good talker or skilled negotiator.

这句话肯定是对的。且平时说话中中很常见