I have experienced this problem as well. I delt with the pain, eventually got a wrist support, and started to contemplate using dumbells or pushup supports. With the help of a friend of mine who is a corrective exercise specialist I have finally addressed the problem, and can do pushups without any pain or extra support. As with most things though, it requires maintenance.
The pain was due to the fact that I rarely asked my wrist to go into that position. As a result my forarms were tight, and the joint was unsupported and weak. As someone who is fit I couldn't tell that by looking, but rather experienced it as pain. As far fetched as this sounds, the problem has gone away by implementing some simple stretches.
Kneel on the floor. Place your hands in front of you on the ground with your fingers aimed towards your knees. Your arms should be straight, and your entire palm flat on the floor, shoulders drawn back and down away from your ears, spine nice and straight. Gently sit back onto your heels until you feel a stretch through your forearms and wrist. Your legs should bear the bulk of the weight while your wrists/forearms experience a targeted stretch. Do this regularly until you can sit all the way back on your heels. Always do it before going into the suspended prone position on the TRX, and before doing regular pushups or planks.
I don't know about you, but I'm not 16 anymore, and it becomes increasingly vital to prepare my body before engaging in activity. In my early 20's I was skeptical, but after learning the hard way I'm done fighting it, and feel a lot better for it.
PS -
you can do a modified version of this by facing the wall. Place your palms flat against the wall with your fingers down toward the ground. The heel of your palm should be level with your shoulders. Again, straight spine, shoulders back and down away from your ears.
Also play with pulling your fingers back from time to time. Eventually you'll regain some of the range of motion you once had.