Hall, the Yankees' 25th-round pick in the 2009 Draft, went eight strong innings, allowing an unearned run on three hits with four strikeouts. He did not walk a batter.
The left-hander hit Dan Cook with one out in the fourth but got Nick Liles to bounce into an inning-ending double play. He faced the minimum before Eliezer Zambrano reached on an error with one out in the sixth and Kyle Mach singled. Zambrano scored on Cook's base hit to lift Augusta into a 1-1 tie.
Hall turned things over to reliever Francisco Rondon after lowering his ERA to 0.77 in three starts. He has given up one earned run on 11 hits over 17 2/3 innings but has not recorded a decision.
Making his South Atlantic League-leading sixth appearance, Rondon (1-2) got a quick out in the ninth but walked Liles and gave up the game-winning blast to Joseph.
The homer was the second of the season and first since Opening day for Joseph, the Giants' second-round pick in last year's Draft.
Augusta starter Kyle Vazquez went seven innings in his second start of the season, allowing a run on three hits while striking out seven. The 21-year-old right-hander has yielded one run on six hits over 13 innings this season.
Reliever Chris Gloor (1-0) picked up the win after striking out Taylor Grote to end the top of the ninth.