Let me clarify first that there are deep conceptual issues with what $E=mc^2$ means, and what it means to verify it. That energy contributes to inertial mass was known before Einstein. In 1900 Poincare showed that electromagnetic field has momentum, hence effectively mass, and implicitly gave it as $E/c^2$. Austrian physicist Hasenöhrl even wrote explicitly $E=3/8\,mc^2$ in a 1904 paper, while German physicist Abraham was interpreted as giving $E=3/4\,mc^2$ (wrong proportionality constants were due to subtle mistakes in calculations).
In 1901-1915 Kaufman and Bucherer conducted a series of measurements of electrons' mass-to-charge ratio, and electrons traveling between 0.25–0.75c indicated an increase in mass in agreement with $E=mc^2$. If we also accept the equivalence of inertial and gravitational mass, bending of the light by the Sun observed by Eddington and Dyson during a solar eclipse of 1919 can be seen as another verification of energy to mass contribution.
However, Einstein's 1905 paper not only extended this effect to all matter, but also boldly claimed the converse, that any mass can be transmuted into (kinetic, and hence any) energy. This is what Cockcroft and Walton are traditionally credited with first experimentally verifying in 1932. The reason it is hard to locate it in their paper is that their intentions were different. They bombarded lithium nucleus by protons to verify that two $\alpha$ particles are produced as a result, not that mass is converted into energy. The rest mass of proton+nucleus was $8.0176$ amu, but of the two $\alpha$ particles only $8.0022$ amu. They noted that the mass defect of $0.0154$ amu “is equivalent to an energy liberation of (14.3 ± 2.7) ×106 Volts” based on $E=mc^2$, which is consistent with about $17$ MeV, the observed value of the kinetic energy of two $\alpha$ particles. However, Cockcroft and Walton also give independent evidence that the products are the $\alpha$ particles, so in hindsight their result can be interpreted as confirming the transmutation of mass into energy. Stanford article has more details and analysis.
Another hindsight verification came from the famous Hahn-Strassmann nuclear fission experiment, where uranium 235 decayed first into xenon 140, and then into barium 140, with some byproducts. The appearence of barium was unexpected because it is so much lighter than uranium, but Lise Meitner in December of 1938 surmised what the reaction was. She also noted the much more sizable mass defect ($0.177427$ amu equivalent to $165$ MeV based on $E=mc^2$, the modern value is $200$ MeV), and predicted massive release of energy, which was observed shortly thereafter. This "splitting of the atom" was a key step towards the nuclear bomb.
The above two experiments are singled out because of precise mass measurements, but in principle conversion of mass into energy was apparent in other early observations. Perhaps the most "philosophically" satisfactory were electron-positron creation and annihilation observed in 1933 by Irène and Frédéric Joliot-Curie using a Wilson chamber in a magnetic field. Since photons have zero rest mass these events signify clean and symmetric conversion of mass into energy and back.
Another example is $\beta$ decay, which led Pauli to conjecture the existence of neutrino in 1930, so named by Fermi in 1933. However, an independent confirmation of neutrino's existence, required to confirm mass to energy conversion, did not come until Cowan-Reines experiment in 1956. Of course, various reactions observed in 1950-60s, when a host of new particles were discovered, also qualify as confirmations. One of the most accurate recent measurements is due to Rainville et al. in 2005, which was explicitly designed to test the mass to energy conversion, it is based on
$\gamma$ ray emission when a nucleus captures a neutron.
