Tjaronn Chery gave the Israeli side hope of a famous victory when he powered home from close range in the 24th minute.
But PSG responded well to that early setback and pulled level before the break when Messi stroked home following fine work by Mbappe.
France international Mbappe put the visitors ahead and Neymar then sealed the three points late on as PSG made it two wins from two in Group H following last week's victory over Juventus.
Josh Cohen denied a clean-through Mbappe after just 70 seconds, while at the other end Gianluigi Donnarumma got down well to keep out Mohammad Abu Fani's strike in the ninth minute.
Maccabi stunned the Ligue 1 giant shortly after the midway point of the first half when Chery stole in at the back post to volley home Dolev Haziza's superb cross from six yards.
PSG restored parity eight minutes before the interval, though, when Messi clipped home from close range after Mbappe's low cross from the left had been deflected into his path.
Frantzdy Pierrot squandered a golden opportunity to put Maccabi ahead again early in the second half, the forward narrowly directing Pierre Cornud's cross past the post.
Messi was denied by a wonderful save from Cohen, but there was nothing the Maccabi goalkeeper could do to keep out Mbappe in the 69th minute as the 23-year-old curled past him after latching onto Messi's pass.
Neymar then put the seal on the victory two minutes from full-time when he latched onto Marco Verratti's ball over the top and fired a crisp finish past Cohen.