Sixth seed Strycova of Slovakia suffered a straight-sets loss to Switzerland's Stefanie Voegele, who claimed a 7-6 (7-4) 7-5 success.
Voegele won the first set that featured four breaks of serve on a tie-break.
Both players continued to struggle on serve in the second as they traded breaks regularly in a contest that lasted two hours and 11 minutes.
But Voegele's fifth break of the match finally settled the encounter in her favour as Wimbledon quarter-finalist Zahlavova-Strycova bowed out at the first hurdle.
Frenchwoman Garcia, seeded eighth for the WTA International tournament, came out on the wrong end of an entertaining encounter with Marina Erakovic as the New Zealander emerged victorious 6-2 4-6 6-4.
Garcia struggled with her second serve, on which she won just 41 per cent of points and was punished for a failure to convert eight of her 10 break-point opportunities.
Erakovic was more clinical as she broke the Garcia serve four times, the last occasion coming in the opening game of the decider.
And the world number 34 was unable to recover from that blow, Erakovic holding firm to seal triumph.
Elsewhere in the draw, Slovakian Anna Schmiedlova overcame Germany's Annika Beck 7-6 (8-6) 6-4.