Depends on how you define the term, of course. The Danish Navy used some of the Madsen Model 1896 semi-auto rifles, later developed into the Madsen machine rifle. The first more-or-less successful combat use of a semiauto was as an aircraft weapon, back in the WWI days when aviators took potshots at each other with handguns. Propellor synchronisation of belt-fed MGs firing through the propellor arc [eventually perfected and technically, initially a semi-auto firing cycle tripped by lobes on the propellor shaft] had not yet come along and the Germans had about enough of the more sporting aspects of aerial warfare. Enter their contribution to that subset of military science, the Swiss SIG-manufactured version of the Mexican Mondragon M1908 semiauto rifle in the 7x57mm Mauser cartridge, known to the Germans as the Flieger-Selbstlader-Karabiner 15.
So far as the first semi-auto weapon to effectively replace the manually operated infantry weapon that preceeded it, that would be Mr. John Garand's wonderful device, the M1 rifle, following initial US Ordnance efforts to turn out a semiauto version of the M1903 Springfield, which included semiauto conversions of the Swiss M1911 straightpull rifle and, eventually, Pedersen's semiauto design that introduced the en-bloc clip-fed magazine of the Garand, initially a 10-round device utilizing Pedersen's .276 cartridge, but very nicely adapted to the .30 M2 and .30 M1 cartridge versions of the .30 calibre cartridge of 1906.