We have studied the effect of oxygen on the time-dependent bifurcations of transient oscillations in the Belousov-Zhabotinsky oscillating chemical reaction in a closed system. Experiments show that oscillations disappear through different bifurcations depending on the oxygen concentration in gas phase above the reaction solution. Oscillations disappear through the time-delayed Hopf bifurcation at low oxygen concentrations, whereas at high oxygen concentrations they disappear through the time-dependent SNIPER (saddle-node infinite period) bifurcation. We propose a kinetic scheme that describes the effects observed in experiments. Good agreement between the experimental data and simulations is obtained.