Michael Laudrup's men had beaten amateur side FC Gravenzande 2-0 on Saturday and followed that up by coming from behind against Sunday's third-tier opponents.
Excelsior seized on a defensive mix-up to take a seventh minute lead, but Roland Lamah equalised on 17 minutes after being played in by Ben Davies.
Lamah then struck the post before Rory Donnelly put Swansea ahead on 28 minutes, and Wayne Routledge was then denied by the woodwork before half-time.
Laudrup changed his entire XI at the interval and had to wait until the 70th minute for the third goal when new signing Alejandro Pozuelo scored.
Michu teed up Kenji Gorre for the fourth goal five minutes later before Pozuelo turned provider to create the fifth goal for Nathan Dyer two minutes from time.