BH Cen is a short-period early-type binary with a period of 0.792(d) in the extremely young star-forming cluster IC 2944. New multi-color CCD photometric light curves in U, B, V, R and I bands are presented and are analyzed by using the Wilson-Devinney code. It is detected that BH Cen is a high-mass-ratio overcontact binary with a fill-out factor of 46.4 % and a mass ratio of 0.89. The derived orbital inclination i is 88.9 degrees, indicating that it is a totally eclipsing binary and the photometric parameters can be determined reliably. By adding new eclipse times, the orbital period changes in the binary are analyzed. It is confirmed that the period of BH Cen shows a long-term increase while it undergoes a cyclic oscillation with an amplitude of A(3) = 0.024 d and a period of P-3 = 50.3 yr. The high mass ratio, overcontact configuration and long-term continuous increase in the orbital period all suggest that BH Cen is in the evolutionary state after the shortest-period stage of Case A mass transfer. The continuous increase in period can be explained by mass transfer from the secondary component to the primary one at a rate of ((M) over dot)(2) = 2.8 x 10(-6) M-circle dot per year. The cyclic change can be plausibly explained by the presence of a third body because both components in the BH Cen system are early-type stars. Its mass is determined to be no less than 2.2 M-circle dot at an orbital separation of about 32.5 AU. Since no third light was found during the photometric solution, it is possible that the third body may be a candidate for a compact object.