They made one change to the starting line-up with Mark Kennedy coming in at full-back, allowing Joe Ledley to move into midfield.
A superb throw-out by Tom Heaton sent Peter Whittingham racing through in the ninth minute and from his corner Darren Purse headed over the Palace bar.
The Bluebirds were attacking with speed and Whittingham fired low but straight at Eagles keeper Darryl Flahavan.
Eddie Johnson had been winning the fans over with his movement up front, but in the 25th minute he blazed the ball out of the ground when well placed just 12 yards out.
In the 31st minute, Chopra was brought down in the area and after a slight pause he rolled the spot-kick into the corner of the net.
Palace equalised six minutes later following a mix-up in the City defence. Flahavan sent a long ball down the middle, Heaton and Kevin McNaughton collided, leaving Sean Scannell the simple task of sliding the loose ball into the empty net.
Early in the second half Ledley went down following a challenge and was only able to return after lengthy treatment.
One minute after returning to the field he put City back in front after racing on to an Eddie Johnson header before turning the ball home from an angle.
Palace manager Neil Warnock made a couple of substitutions and Palace began to push the Bluebirds back on defence.
In the third minute of time added on Eddie Johnson was brought down in the area by Matt Lawrence, who was shown a red card.
Chopra took his second penalty of the game, but this time Flahavan went to his right to push the ball away.
City hung on to the victory but at times they looked a little fragile in defence.