It depends on the context. The phrase in the dark and the word darkness has several meanings. If you want to mean that phrase or words simply absence of light or illumination, both are interchangeable but then, darkness does mean the absence of moral/spiritual values or an unenlightened state.
On the other hand, in the dark also means many things and one of them is to keep someone away from the fact. For instance, Why did you keep me in the dark when you knew that she will not come for sure?
So, again, it's context based. Your sentences:
His face was in darkness can be replaced with in the dark meaning his face was not visible due to absence of light. Also, the second sentence does mean that you want to say an absence of light because the lamp suddenly went.
Please note that if you are using in darkness, it's an adverb as described by WW.
in darkness - without light
So I think both are interchangeable here. But be careful, you cannot replace them always. Refer this...
His holy chants helped us getting out the darkness of the evil spirit --here, if you think about the absence of light/illumination and replace from the dark that way, it won't work.