In English, it's simply saying that nothing will escape the lips (about whatever subject they are speaking about) because they are sealed (locked up, secured) from doing so.
"Oh, he's going to ask Julie out to the prom. Don't tell anyone!"
"My lips are sealed." (Basically saying "I won't tell anyone")
For the Portuguese, it basically says "my mouth is a tomb/grave". Dead people don't talk, so tombs would be silent. It's probably like saying "I'll keep silent" about a certain subject or a secret.
I wouldn't say that "silent as a grave" is that common when spoken. It seems to be pretty common written though (there were many pages on the google search). Even so, if you were to say the sentence "My mouth is a grave" somewhat the literal translation, you'd be looked at with confusion. You could say "I will be silent like a grave" and that would be understood right away, but the most common to me is "my lips are sealed". It might be just where I live, but I'm not even sure I've ever heard anyone say "silent as a grave" except maybe in movies or books... well, and once my ex-wife who speaks Portuguese as a first language said something like "I'll be like a tombstone" which took me a little to find out what she meant, so I think they have a similar idiom in Portuguese.