Schick was dismissed after tangling with Roberts three minutes after the restart, an incident which saw the Swansea wing-back booked.
Roberts collected his second caution 13 minutes from time after catching Tomas Soucek in the face during an aerial challenge, which the West Ham midfielder ensured caught the attention of Romanian referee Ovidiu Hategan.
Wales quickly recovered from that setback as Gareth Bale crossed from the left after 81 minutes, and James’ downward header beat the despairing dive of Czech goalkeeper Tomas Vaclik for his fourth international goal.
The Dragons were seeking their first points in World Cup qualification after losing their Group E opener in Belgium last week.
The Czech Republic – 24 places lower than Wales on the FIFA world rankings at 42 – already had four points on the board after winning 6-2 in Estonia and drawing 1-1 at home to Belgium on Saturday.