The effects of a quadratic f(T) gravity on astronomical observations and solar system experiments were recently considered in previous works. Its deviation from Einstein's general relativity is characterized by a model parameter a and the cosmological constant A whose constraints were respectively found as vertical bar alpha vertical bar <= 2.3 x 10 m(2) and vertical bar Lambda vertical bar <= 1.8 x 10(-43) m(-2) in the solar system. In this paper, a new test of the f(T) gravity by measuring the gravitational time advancement is presented and studied. The gravitational time advancement arises in a two-way light propagation between an observer and a distant spacecraft, where the light traveling time is recorded by the observer's proper time. It is found that (1) relying on their signs, alpha and Lambda can make the gravitational time advancement smaller or larger than the one of general relativity; (2) the configuration of the inferior conjunction between the observer and the spacecraft is more suitable for detecting the advancement, because its effect is almost 3.5 times larger than the one in the superior conjunction; (3) the time advancement could be effectively complementary to Shapiro time delay for gravitational experiments in practice; and (4) the implement of planetary laser ranging and optical clocks in the future will provide much more insight into the f(T) gravity.