Robredo, who reached a career-high ranking of No. 5 in 2006 and is now at No. 72, beat Guillaume Rufin of France, 7-6 (2), 6-4.
The 85th-ranked Kavcic eliminated Moroccan wild card Mehdi Ziadi 6-2, 6-4 and will next face top-seeded Stanislas Wawrinka of Switzerland, the 2010 champion.
Two-time defending champion Pablo Andujar, ranked 56th, opens against a qualifier.
The red-clay tournament is entirely funded by Morocco's King Mohammed VI and is the only ATP World Tour stop in Africa.
Two Moroccans have won the title in their home country: Hicham Arazi in 1997 and Younes El Aynaoui in 2002.