Chopra, signed on loan from Sunderland, fired home a 31st-minute penalty to give City the lead but the Eagles were back on level terms six minutes later through Sean Scannell.
A superb angled strike from Joe Ledley early in the second half regained the lead for the Bluebirds but they had to withstand severe pressure from Palace in the closing stages.
In added time the impressive Johnson was brought down by defender Matt Lawrence, who was sent off, but Chopra's second penalty was pushed away by goalkeeper Darryl Flahavan.
"All victories are hard-earned and this was no different," said Jones.
"Up until they scored we were well on top but it was a bad goal to give away as three of my players lost concentration.
"I am very pleased for Eddie Johnson, who has received a lot of flack since coming to the club. Today we saw him lead the line superbly.
"He won a lot of balls in the air, made good runs and brought others into the play. He has needed time to get up to pace but now we may be able to see the best of him."
Chopra received rapturous applause from the home fans and his determined running brought about the opening goal when he rounded Jose Fonte and cut in down the byline only to be brought down by his pursuer.
After a short delay Chopra struck the spot-kick into the corner to give City the lead their early play deserved.
Palace were level in the 37th minute with route-one football after a Cardiff corner. Flahavan collected and booted the ball downfield where Bluebirds goalkeeper Tom Heaton and defender Kevin McNaughton collided, leaving Scannell the easy task of sliding the ball into an empty net.
Five minutes after the break Cardiff were back in front when Johnson headed on and Ledley fired in from an angle.
The Bluebirds had to withstand tremendous pressure as Palace threw everyone forward but the defence held firm and they should have gone further ahead only for Chopra to miss his second spot-kick.
"I thought we played well and I was very pleased with the performance," said Eagles boss Neil Warnock.
"The home crowd were superb for Cardiff but I don't think the first penalty was deserved but Chopra used everything available to him and good luck to him.
"The disappointing thing for me was that I thought the referee did his best. Referee's should be educated in the game by former players as it is not all about the laws.
"We were slow in picking up Ledley when he came on to the field and that proved costly."