The Gamecocks (8-0) trailed 6-1 in the fifth inning when the comeback began with Greiner's shot to left with the bases full. An inning later, Greiner drove in the go-ahead run with a two-out single deep in the hole at short that Tyler Krieger couldn't handle.
Relievers Cody Mincey and Joel Seddon held Clemson to one hit the final 3⅔ innings.
The Gamecocks have won 11 of the past 14 games with Clemson (6-2), including six in a row at Carolina Stadium.
Greiner’s blast was his third homer of the year and South Carolina’s first grand slam since Greiner did it in April 2012 against Citadel.
The teams play in Greenville today before finishing the series at Clemson on Sunday.
The likely pitching matchup today is between two sophomores who have started 2-0, South Carolina’s Jack Wynkoop and Clemson’s Matthew Crownover.
The finale will feature Gamecocks freshman standout Will Crowe, who has thrown 12⅔ scoreless innings and has yet to surrender a run in his first two college starts.
Clemson entered with a .322 team batting average, second in the Atlantic Coast Conference, and is looking to win a season series against its rival for the first time since 2006.