The first place team in the National Association of Professional Base Ball Players, the Boston Red Stockings stopped by the Capitoline Grounds in Brooklyn to take on one of the NA entries there, the Atlantic Club. According to the New York Times about 200 people showed up and it wasn't much of a contest as Boston walloped the Atlantics, 23-3.
The accounts of the game are pretty sketchy. I do know that Boston scored in every inning but the seventh and led 18-3 after six innings. Harry Schafer, the left fielder this day, but usually the third baseman, had five hits and scored four runs. Dave Birdsall was the catcher and batted fourth. Birdsall was Boston's lone reserve player and he got into only 16 games. Boston used just 10 players during its 48-game season, during which it went 39-8-1 and easily claimed the pennant. The 48-game season for Boston started on April 30 and finished on October 22.
The Atlantics went 9-28 and lost their first nine games. They didn't win a game until July 2. And they kept playing off and on until October 31. The Atlantics featured two players of note. One was third baseman Bob Ferguson, forever immortalized in history books with the nickname "Death to Flying Things". He played for 14 seasons in the majors and also served as an umpire and manager. The other star was shortstop Jack Burdock, who played for 18 seasons and would manage Boston's NL entry in 1883 for half the season. Burdock would play most of his career at second base.
The Times game story mentions Ferguson making a one-handed catch of a liner off the bat of Boston's Charlie Gould and then having to retire from the game in favor of someone named Jackson. However, the mysterious Mr. Jackson didn't show up in the boxscore, although Baseball-reference.com does credit a Sam Jackson as appearing in four games for the Atlantics in 1872.
The last surviving player from this game 134 years ago was Boston's Hall of Fame shortstop George Wright, who didn't pass away until 1937 at the age of 90.
As for scoring 20+ runs, Boston would do so eight times during the 1872 season, three times against the Atlantics.
Sources: Retrosheet, New York Times, Baseball-Reference.com