Getting an explosion in absence of oxygen is no big trouble.
The first obvious way is to use an explosive that already has its oxygen stored within. Pretty much everything used in present-day guns or cannons qualifies for that, this is why you can (in principle, neglecting other little details) fire a gun underwater or even in space no problemo. Pretty much every solid rocket fuel also qualifies.
Anything and everything used by and used against submarines during World War II would demonstrably do. Torpedoes, water bombs, and mines explode just fine without oxygen.
The second obvious way is to use a pressurized tank full of hydrogen and a pressurized tank full of oxygen, break the tanks open, add the tiniest little spark, and you have hell break loose. Gas explosions, next to dust/aerosol explosions ("thermobaric weapon") are some of the most devastating explosions we know.
If you wish to be a bit more posh, substitute oxygen with chloride, and you do not even need a spark. Releasing the gas and exposing it to sunlight suffices. That's truly an explosion in absence of oxygen.
The third, somewhat cheating way is to not cause an explosion at all, but to have gas expand rapidly anyway -- to the exact same effect. How would you do that? Take a chemical which is cheap/easily produced or abundant, and has a low boiling temperature (say, carbon dioxide, but quicksilver or water would do as well) and heat it to considerable temperature (2,000-3,000K) very rapidly. I'll leave to your imagination what the best way would be to achieve this. If nothing else, it might convert kinetic energy from the impact (but in Sci-Fi, it can be whatever... energy from a "plasma generator"?).
This is not-an-explosion, which however has the same net effect and works perfectly well in absence of oxygen.
Nuclear weapons are out of the question. But why? The rare element catalysium will, if added to normal plutonium, increase the yield to upwards of 97%, and with a proper bounconium reflector to upwards of 99.975%. Fallout is minimal.
Bullshit? Well, no, it's actually exactly what, in the real world, boosted fission bombs (and beryllium or carbide reflectors) are trying to achieve, and they do achieve it, only not at that scale (but it's fiction, after all).