"That binds each living thing?" Nah - sounds kinky. "Links?" Maybe too mechanical. I dunno - it didn't bother me the way it was. You could use "springs" as a verb:
That magic is the power
And from Life's own (something - "well?") it springs

The magic, to me, is what happens when people believe they can do something extraordinary and things just come together in the right place, the right time. The most magical thing about such an occurrence is that people DO believe enough to make it happen.

Funny - your song reminds me very much of one I wrote. The meter is even very close:
When we sing a spark within us wakes and is fanned into a flame
To those who hear our spirit's voice, it calls in them the same
And soon that light within us grows and wells up like a spring
And it fills the barren emptiness, and it's why we sing.