Learn Hardcore Chinese: xìngkuī màikèfēng kěyǐ yòng, bùrán wǒmen bùnéng chàng kǎlāOK. - Fortunately the microphone worked; otherwise we could not sing karaoke.

A very common learner question is how it differs from 幸好. The two are often similar, and in many everyday sentences they can both be translated as luckily.

A useful difference:

幸亏 often sounds a bit more like thank goodness that..., sometimes with a feeling that a bad result was narrowly avoided.
幸好 is a little broader and often more neutral.

So here, 幸亏 fits well because the second half says there would have been a problem otherwise.

Why is 幸亏 placed at the beginning of the sentence?

Putting 幸亏 at the beginning sets up the whole situation: Luckily, ...; otherwise, ...

So the sentence structure is:

幸亏 + good thing
不然 + bad result

That makes the logic very clear:

幸亏麦克风可以用
不然我们不能唱卡拉OK

In natural English, this is like:

Luckily the microphone worked, otherwise we wouldn’t have been able to sing karaoke.

You can think of 幸亏 as a sentence-level adverbial phrase introducing the speaker’s relief.

What does 麦克风可以用 mean exactly? Is it can use the microphone or the microphone works?