England under-21 international Abraham, signed on a season's loan from Chelsea, headed home in the 80th minute.
Paul Clement's side - without injured Fernando Llorente and transfer-target Gylfi Sigurdsson - took a first half lead through Oliver McBurnie.
But substitute Franck Tayou equalised on 69 minutes for the Kickers before Abraham had the last word.
Manager Paul Clement told BBC Radio Wales: "I was satisfied with the way we played, lots of players got minutes under their belts.
"Half the team played 60 minutes, the other half played 30. Lots of things to work on, but with three and a half weeks to go to the start of the season, we are moving in the right direction."
Swansea drew 2-2 with Philadelphia Union in the first game of their tour, having previously lost a friendly 1-0 at Barnet.
They complete their American training trip against North Carolina, on Saturday.
Tammy Abraham scored a winning goal on his debut as Swansea City beat Richmond Kickers in the second USA tour game.