Lavoisier was writing before relativity was known and his statement is wrong: mass is not conserved when relativistic effects are non-negligible. You cannot use his statement to imply that total energy is conserved.
The conservation of energy is due to a fundamental symmetry called time shift invariance. Noether's theorem tells us that this symmetry is linked a conserved quantity, and in the case of time shift invariance the conserved quantity is energy.
The equation $E=mc^2$ is a special case that applies only when the object with the mass $m$ is stationary. The full equation is:
$$ E^2 = p^2 c^2 + m^2 c^4 $$
where $p$ is the relativistic momentum. This reduces to $E=mc^2$ when $p=0$ i.e. when the mass $m$ is stationary.