Yes. In chemistry, the general term is "redox reaction" where one subastance is reduced while another is simultaneously oxidized. With fire, oxygen is reduced, while carbon, hydrogen, sulfer, etc are oxidized to produce compounds like CO, CO2, H2O, SO2, etc.

Other redox reactions that you might consider similar to burning without oxygen use other reactive oxidizers like Chlorine, Bromine, Fluorine, etc.