Major League Baseball's official record book states that Pete Rose got his record-breaking 4,192nd career hit off Eric Show of the Padres, a single in the first inning at Riverfront Stadium in Cincinnati September 11, 1985.
This official "fact" is premised on the official record of MLB, in which Ty Cobb has 4,191 career hits. ("Fact" is in quotes for a reason, which will become evident shortly.)
It's become generally accepted among baseball researchers that Cobb's actual hit total was 4,189. His baseball-reference page says so. Fangraphs says so. Baseball Almanac says so.