With Neymar, again the subject of transfer speculation with reports he is "begging" for a return to Barcelona, left out of the squad after Thomas Tuchel described him as "very tired", PSG found it difficult to break down the league's fourth-bottom side until the 57th minute.
Cavani broke the deadlock from the penalty spot after Alexis Blin was penalised for handball and, following a red card for Amiens' centre-back Khaled Adenon, the Uruguay international teed up Mbappe before Marquinhos added a third as PSG stretched their lead at the top back to 13 points.
A well-worked eighth-minute corner from PSG saw Julian Draxler denied by the right leg of Regis Gurtner, who enjoyed a largely comfortable first half despite PSG having over 75 per cent possession.
Mbappe could only produce a tame poked finish from Marco Verratti's throughball, though more testing was Angel Di Maria's fierce effort that Gurtner tipped over as Mbappe tried to claim what would have been a controversial penalty after going down off the ball in the build-up.
For all PSG's clear superiority, they would have trailed at half-time if not for a superb diving save from Alphonse Areola to thwart Eddy Gnahore at the end of an Amiens counter-attack.
All Amiens' good work in the first half was undone 11 minutes into the second as Alexis Blin handled as he dove in to block a Di Maria effort and Cavani coolly sent Gurtner the wrong way from 12 yards.
Their task was made even harder when Adenon received a second bookable offence and his marching orders for a poor challenge on Mbappe, who was soon celebrating as he tucked away a confident close-range finish from Cavani's fine right-wing cross.
Substitute Moussa Diaby saw an effort cleared off the line before Marquinhos made the scoreline more flattering by lifting into the net 11 minutes from time.
14 - Kylian Mbappé has scored 14 goals in Ligue 1 this season, only Lionel Messi (16) has scored more in the Top 5 European leagues. He is one goal away from his best tally in a top-flight season (15 with Monaco in 2016/17). Diamond. @PSG_English #ASCPSG pic.twitter.com/T15QiGpY5g— OptaJean (@OptaJean) January 12, 2019
What does it mean? Amiens make it easy for below-par PSG
The scoreline certainly does not tell the whole story for PSG, who were uninspiring before referee Florent Batta pointed to the spot. Once Adenon saw red the result was never in doubt, but Tuchel's team will need to improve significantly if they are to challenge for the Champions League, with Manchester United awaiting them in a month's time.
Verratti pulls the strings
As PSG laboured in the opening 45 minutes, the one player whose performance merited a goal was Italy midfielder Verratti, who consistently exploited the gaps in the Amiens defence with his wonderful range of passing.
Adenon red ends Amiens hopes
Amiens had shown enough to suggest they could have at least a slim chance of coming back after Cavani's penalty, but Adenon's rash tackle saw any prospect of a surprise result disappear. They remain 17th after seeing a run of three games unbeaten in the league come to an expected end.
PSG welcome their cup conquerors Guingamp to the Parc des Princes again on Saturday, while Amiens make the short trip to neighbours Lille a day earlier.