the conservation of energy is violated in cosmology - in all situations described by general relativity in which the time-translational symmetry of the "background" is broken. That's clearly the case of the Big Bang, too.
Again: By Noether's theorem, the conservation of energy is linked to the time-translational symmetry (the properties of the Universe don't depend on time) which is broken in an expanding Universe.
So one can see that the total energy/mass of the Universe is not conserved in time. In particular, the "dust" with zero pressure has a conserved total energy/mass. However, the energy carried by radiation - such as photons - decreases as $1/a$ - inverse linear size of the Universe - because the wavelength gets larger as well, which decreases the energy of each quantum.
On the contrary, the energy/mass carried by the dark energy is increasing with the volume because the energy density is constant - that's why dark energy is the normal realization is known as the cosmological constant. The density is constant but the volume of space is growing: the total energy is growing, too.
It is very likely that the huge mass of the Universe around us was created by inflation. During inflation, there was also a nonzero "dark energy" - energy density of the vacuum - which was constant while the volume of space was exponentially growing. This created lots of energy, and at the very end, a big part of the energy (kinetic energy of the inflaton which is proportional to the total energy etc.) was converted to ordinary particles that eventually gave rise to the galaxies etc.
So the exponentially large mass of the Universe is a large, unnatural number, but this unnatural number is actually explained by inflation. As Alan Guth, the main father of inflation, said, the Universe is the ultimate free lunch.