Blake bested Spain's Albert Ramos 6-4, 6-3, firing seven aces en route to the win on the hard courts at Wake Forest University.
Goffin, meanwhile, beat France's Nicolas Mahut 7-6 (7-3), 6-1, while Darcis defeated countryman Xavier Malisse 6-3, 6-0.
Also notching first-round wins were American Ryan Harrison, Santiago Giraldo of Colombia and Russian Alex Bogomolov Jr., who dispatched Tunisia's Malek Jaziri 6-2, 6-2.
The Giudices celebrate Easter as they prepare for jail
Justin Bieber crashes Drake Bell's album release party