Schiavone, whose last tour title came at the event 12 months ago, was completely outclassed on Tuesday in a 6-3 6-2 defeat at the hands of qualifier Beatriz Garcia Vidagany, who progressed to round two at the WTA international clay-court event in Marrakech.
The Italian struggled on serve throughout the match, winning just 19 of 52 points, and Garcia Vidagany took full advantage, claiming seven breaks.
Second seed Svitolina was also a first-round loser, with Austrian Patricia Mayr-Achleitner beating her 7-6 (7-2) 3-6 6-2 in two hours and 19 minutes.
The match was riddled with breaks, but Mayr-Achleitner made eight - compared to the seven of Svitolina - and that was enough to give her victory.
Fourth seed Bojana Jovanovski was also a surprise casualty - Spain's Maria-Teresa Torro-Flor accounting for her exit in a straightforward 6-2 6-2 success.
Top seed Daniela Hantuchova, handed a wildcard for the tournament, had to come from a set down to beat Czech qualifier Renata Voracova 3-6 6-2 7-5.
The 30-year-old started sluggishly and dropped serve on three occasions in the opening set, but bounced back strongly to defeat her opponent in two hours and 12 minutes.
Seeds Garbine Muguruza and Peng Shuai won well, with the former dispatching Julia Glushko 6-2 6-2. Peng overcame wildcard Ons Jabeur 6-4 2-6 6-1.
Elsewhere, there were straight-sets victories for Alison Van Uytvanck, Maryna Zanevska, Stefanie Voegele, Estrella Cabeza Candela and qualifier Lara Arruabarrena, while Romina Oprandi needed three sets to defeat Anastasia Grymalska.