An FA Cup finalist in his Palace playing days, Pardew won only once in the competition across four seasons at previous club Newcastle United, but was given little to worry about on Sunday as his new side brushed past their non-League opponents.
The visitors took the lead after 10 minutes when Wilfried Zaha's cross was guided home by Dann at the far post, and the centre-back doubled the advantage before half-time.
Dwight Gayle added a third midway through the second half with a near-post drive with Kevin Doyle's first Palace goal capping off a highly satisfactory afternoon's work at a packed Crabble Athletic Ground for the Premier League strugglers.
However, it is the end of the road for Dover, who managed just a solitary shot on Wayne Hennessey's goal and bow out at the third round for the second time in their history.
Some 86 places separated the two teams coming into this contest, and it was unsurprisingly the top-flight outfit that asked most of the questions during the opening exchanges, despite Pardew making six changes to the side.
A jinking run from Zaha found space down the left, before curling a cross for Glenn Murray at the far post, whose header was palmed away by goalkeeper Andy Rafferty, the striker - recently returned from a loan spell at Reading - also seeing two more chances go begging.
The inevitable breakthrough came on 10 minutes, when a teasing cross from the lively Zaha was flicked by a Dover head into the path of Dann, who volleyed home at the far post.
An unchanged Dover came into the tie on the back of a 16-match unbeaten run in all competitions, but struggled for possession on a rough pitch that proved as much to their own disadvantage as their Premier League opponents.
And things were made even harder for the hosts 12 minutes before the break as Dann rose highest to meet Barry Bannan's corner and power home a towering header to double Palace's lead.
The visitors looked in no rush to add to their lead after the break, although Stuart O'Keefe should have kept his shot down on 62 minutes, before some desperate defending saw Gayle's shot blocked soon after.
Gayle was not to be denied though, and found the net on 68 minutes, slaloming past Connor Essam on the left, before cutting inside the area and beating Rafferty at his near post.
The win was rounded off three minutes from time, when substitute Jerome Thomas' cross found substitute Doyle, allowing the Irishman to open his account from close range and cap a happy reunion for Pardew and Palace.
