The Rams were knocked out of the competition by Crawley last year but Robinson ensured they avoided defeat this afternoon.
He needed two bites at the cherry to put the hosts in front following Jamie Ward's cross and although Derby dominated they could not kill the game off.
The Eagles, without several first-team players, came on strong in the closing stages as they attempted to find an undeserved equaliser but they were second best throughout.
It was perhaps an indication of how seriously the sides were taking the competition, with Derby fielding the same line-up from the 1-0 victory over Hull on Bank Holiday Monday.
Palace boss Dougie Freedman, with one eye on Tuesday's League Cup semi-final first leg clash with Cardiff, made seven changes to the side that lost to Leicester last time out.
Robinson was Derby's hero at the KC Stadium and he was on target once again as Nigel Clough's men took a early lead.
Ward's cross on the left was flicked on to Robinson, who slotted home at the second attempt after Palace goalkeeper Lewis Price had parried his initial close range effort.
Buoyed by his eighth goal of the season, Robinson nearly doubled his and Derby's tally immediately after but his 25-yard strike produced a fine save from Price.
Ward then missed the target by a whisker from a similar distance, while another long-range effort from the striker was fumbled by Price but no one could capitalise.
Palace were struggling to get to grips with their task although Antonio Pedroza could have given them an undeserved equaliser in between Ward's efforts but his strike in the area narrowly missed the right hand post.
It was one of precious few chances for them in the opening 45 minutes and Derby ended the half strongly, with Callum Ball culpable of missing the target from close range.
Ball was at the forefront of the action once again after the interval when he collected the ball from Robinson, but after being given time and space to pick his spot, he scuffed his effort to give Price a comfortable save.
Derby looked the most likely to extend their lead and Craig Bryson nearly went clean through but he could not quite latch on to Robinson's through ball.
Price had to be at his best to tip over Paul Green's header at the back post from Ward's corner.
Despite being just one point and two places behind Derby in the Championship, Palace were being outclassed although their defence were put under pressure.
And while there was just one goal separating the sides, Palace were always likely to grow in hope although Calvin Andrew's header over the bar was the closest they came to forcing a replay.