The 22-year-old Mbappe netted the opener with less than two minutes played before adding to his tally with a thumping volley just five minutes later as PSG came out of the blocks flying.
Mbappe added an assist later in the first half, nutmegging his marker on the left flank before switching the play to Messi in the centre and the Ballon d'Or winner lashed home from the edge of the box.
The visitor scored a consolation through Mats Rits in the 68th minute, but PSG regained its three-goal advantage just eight minutes after conceding as Messi rifled home a penalty.
PSG wasted no time in opening the scoring, with Mbappe curling in off the right post after Simon Mignolet failed to deal with Nuno Mendes' left-wing cross, parrying it back into a dangerous area.
Mbappe struck again in just the seventh minute with an emphatic volley after a delightful dink over the Club Brugge defence by Angel Di Maria.
The French striker turned provider in the 38th minute, bursting away down the wing before finding Messi, who curled a fantastic strike inside the left post from just outside the area to make it a 3-0 lead.
Club Brugge pulled one back midway through the second half as Noa Lang darted into space from the left and squared to Rits, who curled a low effort into the bottom-left corner despite a touch from Gianluigi Donnarumma.
However, Messi drew a foul from substitute Ignace Van der Brempt in the area and stepped up to drill a superb penalty into the bottom-right corner that left Mignolet rooted to the spot and rounded off the scoring.