How did the cardiovascular system evolve?
While others have addressed the big picture aspects of your question, I think it would be useful to look at the specifics. 

Have a look at the heart (or more accurately, the hearts) of the earthworm:  

They're nothing more than veins with some pumping muscles wrapped around them. It seems almost a stretch to call them hearts, they are shaped so different from what we think of as a heart proper.

Also, note the earthworm's lungs, or rather, lack of them. It doesn't have any! Why not? It doesn't need them. It gets enough oxygen through its skin via osmosis. It's only larger organisms that need dedicated systems to concentrate oxygen from the surrounding environment.

So, the worm has a simpler system (no chambered heart, no lungs) that works.

All vertebrates descended from a common ancestor that was very similar to this earthworm. It had simple hearts, and no lungs. You can follow the evolution of the human heart through fish heart: 

which is a more sophisticated pumping vessel with two chambers.

Amphibians evolved from fish, reptiles from amphibians, and mammals from reptiles. In this diagram, you will find that the heart becomes more sophisticated and efficient in each:


So, this should give you a good idea of the evolution of the human heart from simpler, working system. I won't take the time to draw out the evolution of blood vessels or lungs; maybe someone else will, or you can google them yourself, the information is readily out there. But they all follow the same pattern: gradual, incremental improvements on working, simpler systems.