Well, the answer can be kind of complex, but the reason why newer jet aircraft or more effecient is primarily because of how they produce thrust. The engines on newer jets are extremely efficient, many orders of magnitude greater than the old style turbojets.
Mostly, they use less fuel because they are "turbofans" and not turbojets. The burning of fuel and compresed air inside the engine creates a high velocity, high temperature gas which is ducted toward the rear of the engine. This high energy air crosses a series of trubines (sort of like a water wheel, but for high temperature/velocity gases). These wheels, among other things, spin via a shaft a huge fan in front of the engine. This fan in turn greatly increases the speed of the air coming in the front of the engine and accelerates it around the core of the engine (where all the combustion takes place) and out the back of the cowling. This high velocity flow represents a large portion of the thrust developed by a turbofan engine. The reason this is more efficient in comparison to older turbojets (where all the air that goes in the front of the engine is used in the combustion process) is that only a small portion of the air going through the front of the turbofan engine is used in the burning of fuel. Thus, the less air going in the combustion chamber, the less fuel is needed.
This also has the added benefit of putting a shroud of cool air around the hot exhuast gases from the core, having a noise supressing effect.
You are also correct about lighter materials... composite structures do have allot to do with the fuel savings as well.
Sorry for the long winded narative... help it helped.