Peter Whittingham looked to have Cardiff cruising, before Steve Elliott set up a grandstand finish when he reduced the deficit after 75 minutes.
But substitute Josh Magennis headed home late on to ensure there would be no upset as the League One side bowed out in the second round.
Bluebirds boss Jones said: "We had to be at our best - it doesn't matter who we play. We didn't get our rewards in the first half. A bit more care and luck would have won it.
"We allowed them back into the game by giving away a sloppy goal. But I was really pleased with our movement and passing.
"We've got a strong squad here and wanted to use them. There's a good buzz about the place; everyone is enjoying themselves, and it was important to continue the winning streak."
Chopra had spurned a hat-trick of chances, before he finally broke the deadlock in the 38th minute.
Kelvin Etuhu robbed Carl Regan and he fed Chopra, who did the rest as he curled his shot beyond Rovers goalkeeper Rhys Evans.
The lively Etuhu was denied twice, before Whittingham looked to have ended the tie in the 68th minute.
Evans could only palm Chris Burke's cross into the path of Whittingham, and he slotted home the loose ball from close-range.
But Elliott headed home Stuart Campbell's corner in the 75th minute.
Campbell missed a chance to send the tie into extra time when he fired over the bar after a flowing move, with time running out.
Magennis settled any nerves for Cardiff when he nodded home with just four minutes remaining.
Rovers manager Paul Trollope said: "We were disappointed with our first-half display.
"They ripped us apart on a number of occasions. Their movement behind us was fantastic. We spoke about a few issues at half-time and asked for a response. They did that and got on top for spells.
"We couldn't quite get an equaliser, but we got a foothold on the game and had a small window to force extra time.
"It wasn't to be. But there are lessons we could learn from playing against a good Championship side.
"The quality was better than we see week in week out. We accept the defeat and move on now."