"Time needed to complete a work" -- is this correct?

Both are grammatically correct.

However, in my experience, a work is generally used to describe a piece of art.

It depends on what you’re trying to say, but it’s probably not correct.

First the more common situation: if you are (for instance) a student doing essays, or a mechanic working on cars, then the phrase you gave would not be correct. Instead, you could say something like

time needed to complete work

or

time needed to complete a task

or …this work, …the work, …my work, etc., depending on context. You can have one task but you can't have one 'work' (that sounds really wrong like 'one water').

A less common (but perfectly OK usage), 'work' as a count noun is typically only used for works of art. So if you are an artist explaining how long it takes you to finish a painting, then yes, you could talk about time needed to complete a work.

But that phrase standing alone without context sounds wrong.