Darren Ward fired home from close range after Danny Butterfield's right-wing cross to put Palace ahead.
Matt Heath nodded in to level for the home side but Clinton Morrison's stooping header put Palace back ahead.
Ben Watson scored with a low drive for the visitors' third and, after Claus Jorgensen was sent off, Jon Macken converted an Andy Johnson cross.
Crystal Palace manager Iain Dowie:
"No disrespect to Coventry but I think there is a little bit more to come from my team.
"But I am not going to start making predictions about the season, I don't like doing that.
"We have to focus on our next game and make sure we are ready, but we are pleased with the result. We created lots of chances in both halves."
Coventry: Bywater, Duffy, Shaw, Heath, Hall, Nalis, Doyle, Hughes (Sofiane 45), McSheffrey, Adebola (Morrell 79), Scowcroft (Jorgensen 68).
Subs Not Used: Whing, Flood.
Sent Off: Jorgensen (82).
Goals: Heath 45.
Crystal Palace: Kiraly, Butterfield, Fitz Hall, Ward, Boyce, McAnuff, Watson (Hughes 62), Soares, Reich, Freedman (Macken 87), Morrison (Johnson 79). Subs Not Used: Speroni, Hudson.
Goals: Ward 14, Morrison 45, Watson 58, Macken 90.
Ref: A Wiley (Staffordshire).