In complex analysis (and complex geometry), an "entire function" usually refers to a complex-valued function. Since the reciprocal mapping $f(z) = 1/z$ (extended to the Riemann sphere/complex projective line) is not complex-valued ($f(0) = \infty$ is not a complex number), one doesn't usually call $f$ "entire".
This mapping $f$ is, however, a (global) holomorphic mapping of the sphere to itself, in fact, a projective automorphism (a holomorphic bijection with holomorphic inverse). In this sense, your intuition is perfectly correct.
In the same sense, every rational function in one variable defines a global holomorphic mapping of the sphere to itself, and the rational functions of degree one (i.e., Möbius transformations) are precisely the holomorphic automorphisms of the sphere.