The spontaneous injection of a polymer chain into a spherical cavity from a narrow cylindrical tube is investigated by using Monte Carlo simulation. A continuous phase transition from a partially injected state to a completely injected state is found at the critical radius (R-C) of the spherical cavity. The dependence of R-C on the polymer length (N) and the radius of tube (r) can be described by a scaling relation R-C proportional to N-1/3 r(1-1/3 nu) with nu as the Flory exponent. In the partially injected phase, the number of injected monomers (m(in)) in the sphere can be written as m(in) proportional to R(3)r((1-3 nu)/nu) in good agreement with the results from the free energy landscape. The injection time (tau) for the polymer ejecting from the tube is a function of R and behaves differently for the partially injected phase and the completely injected phase, which can be described qualitatively by a simple kinetic model.