The Americans began strongly, winning eight straight points in their opening two service games before Lisicki had to stave off a couple of break points on serve in the fourth game.
Bryan then saved three break points on his serve before the third seeds broke Kas's serve and went on to take the set 6-3.
Raymond and Bryan threatened to break Lisicki's serve early in the second but the young German responded and just held on.
The set then stayed with serve until the tenth game when the Germans earned a set point on Bryan's serve and converted to send the match into a 'super' ten-point tie-break.
The Americans were rampant in the breaker, racing to a 7-0 lead before moving to 9-2 and seven bronze-medal points.
They took the third as Kas netted a backhand to see the tie-break end 10-4.