In science, the conservation of mass-energy or matter-energy states that the total amount of mass (or matter) plus energy in a system, or sometimes it is said the the universe, is constant. 
In 1896, the concept of the of conservation of mass-energy is said to have entered into the corpus of science following firstly French scientist Antoine Becquerel’s 1896 discovery of radioactivity in uranium, in which energy seemed to be coming from nowhere, and second German-born American physicist Albert Einstein’s 1905 formulation of the mass energy equivalence formula.  There does not seem to be, however, any specific publication that purports the law of conservation of mass-energy as a new scientific principle of science, but rather this seems to be a urban version of blend of the conservation of mass and the conservation of energy.
In 1983, chemists Robert Smoot and Jack Price stated that: 
“Scientist [believe] that the total amount of matter and the total amount of energy in the universe [are] each constant. These laws are the conservation of energy and the conservation of mass. Thus we see that the two conservation laws are really just one law. This law is known as the law of conservation of matter-energy.”
The statement of there being a law of conservation of matter-energy, to clarify, is a non-scientific verbalized aggregation of several principles: conservation of mass (Lavoisier, c. 1780), conservation of energy (Joule, 1843), first law of thermodynamics (Clausius, 1850), mass-energy equivalence (Einstein, 1905), intertwined with references to an ill-defined type of 'matter-energy', such as used in discussing proton-proton chain reactions (Eddington, c. 1920), confused with particle physics (c. 1980s) descriptions of the fundamental matter of the universe, defined as types of fermions, and the energy of the universe defined as types bosons. Subsequently, in an oversimplified way, this set of principles is sometimes referred to, in a colloquial, non-rigorous scientific way, as the law of conservation of matter-energy (or conservation of mass-energy).
To give an example, in stellar proton-proton chain reactions, hydrogen atoms transform into helium atoms and in the process part of the hydrogen mass is converted into energy. Specifically, in each reaction set, comparing the mass of the final helium-4 atom with the masses of the four initial protons, reveals that 0.007 or 0.7% of the mass of the original protons has been lost. This mass has been converted into energy, in the form of gamma rays and neutrinos released during each of the individual reactions. The total energy produced in one whole chain is 26.73 MeV. The exact details of these types of particle physics reactions, however, tend to become lost in translation when simplified into simpler science.
In 2003, authors Charles Carraher and Raymond Seymour declared: 
“The universe is composed of matter/energy and space. Matter/energy is conserved as described by the law of conservation of matter/energy.”
No one, however, has actually formally proposed a ‘law of conservation of matter-energy’.
1. Christensen, John W. (2003). Global Science, Energy, Resource, Environment (pg. 82). Kendall Hunt.
2. Smoot, Robert C. and Price, Jack. (1983). Chemistry: a Modern Course (pg. 11). Charles E. Merrill.
3. Carraher, Charles E. and Seymour, Raymond B. (2003). Giant Molecules: Essential Materials for Everyday Living and Problem Solving (pg. 5). Wiley-IEEE.
4. Mills, David. (2006). Atheist Universe (section: The Conservation of Mass-Energy, pgs. 72-74). Ulysses Press.
● Conservation mass-energy – ChemTeam.info.