This video was posted on the game's sub-Reddit yesterday (starts at 0:41, all the pertinent info is covered in 90 seconds):
Coolness explained
To sum it up: There's a hidden stat called "coolness" which is affected by clearing rooms without taking damage. Once your coolness level reaches a certain point, you'll receive a drop (ammo, key, heart, etc.). Your "coolness" won't decrease or reset if you take damage in a room, it'll just take longer to reach that threshold. So the more rooms you clear flawlessly, the more frequently you'll see drops.
Also, supposedly you won't receive money from any enemy that manages to hit you.
Sounds like you're talking about this attack (0:59 in):
Dragun gun attack
Moving to the bottom-center of the room and jumping forward diagonally should do the trick.