Michael
Good question.
I'll admit that I'm very prejudiced, being a former Hi-Power shooter and current collector of US National Match ammunition.
I'm sure there were cartridges during the last years of the 19th Century capable of meeting and exceeding the sub-1 MOA test. Whether they were able to do it consistently, at all distances, is another thing. But, as early as 1908, the new Cal .30 Model 1906 cartridge and Springfield M1903 Rifle were doing it both consistently and at long range. Early lots of PALMA and Olympic ammunition tested at Frankford Arsenal hovered right around the 1 MOA mark. For example, the 1908 Olympic ammunition was officially recorded at .9 MOA at 600 yards. Standard National Match ammunition lagged behind, but not by far. By 1921 the famous "Tin Can" Match ammunition was tested right at 1 MOA at 600 yards and after that year most NM ammunition held under 1 MOA. The 1924 National Match tested at .75 MOA at 600 yards, a record that would stand until 1962.
Of course, by the 1930s there were many small caliber wildcats capable of besting the Cal .30, at the shorter distances. But for consistency I'd say it was the first.
As usual, this is JMHO. Maybe German will chime in with his opinion.
Ray