The Premier League strugglers led 2-0 at half-time thanks to a brace from centre-back Scott Dann, the first time the 27-year-old had ever scored two goals in a match.
And Palace completed a satisfying afternoon’s work by netting twice more after the interval, Dwight Gayle finding the target before substitute Kevin Doyle completed the scoring with a strike three minutes from time.
The visitors could have been in front as early as the first minute through striker Glenn Murray, who recently returned to the south-east London outfit following the end of his six-month loan spell at Championship side Reading.
However, Dover goalkeeper Andy Rafferty did brilliantly to keep Murray’s close-range header out and the shot stopper was at it again just moments later to deny the 31-year-old on two more occasions in a matter of seconds.
As it turned out, those misses proved to be just a stay of execution for the Conference side, with Palace taking the lead after 10 minutes following some good approach play by wide man Wilfried Zaha.
The winger’s trickery crucially earned him some space on the left flank, before his centre was flicked straight into Dann’s path at the back post by a Dover player, with the defender side-footing home from close range.
And it was Dann who was again in the right place at the right time to head the visitors into a 2-0 lead 11 minutes before the break following a Barry Bannan corner from the right.
Indeed, the Premier League side could have stretched their lead even further before half-time, only for Rafferty to react smartly at his near post to keep out a deflected drive from right-back Joel Ward.
But with the Conference side, who sit 86 places below their opponents in the English football pyramid, struggling to barely get out of their own half at times, the second period was merely a question of just how many goals Palace could score.
And Pardew would have been delighted to see his new charges, who had gone into this tie having found the back of the net on only two occasions in their previous seven games, score twice more after the break.
Their third was netted by Gayle, who showed some neat footwork in the Dover box to make space for himself before beating Rafferty at his near post with a fiercely-struck effort, and then Doyle rounded off the scoring late on.
Man-of-the-match Bannan firstly picked out substitute Jerome Thomas with an inch-perfect pass, and the pacy winger’s cross was tapped in from just six yards out by the Republic of Ireland international.
It was Palace’s first win since they beat Liverpool in the Premier League in November, and just a fifth in total in 23 matches in all competitions this season.