Mbappe was the first to strike and Messi had the France forward to thank for his second-half double after the Ligue 1 side had fallen 2-1 behind following goals by Andre Silva and Nordi Mukiele either side of the interval.
Mbappe, who missed a stoppage-time penalty, set up Messi for the equaliser and won the spot kick that the Argentine converted with an ice-cool chip after what had been a lacklustre performance from the club’s new signing.
The result left PSG top of Group A with seven points from three games, one point ahead of Manchester City, which demolished Club Brugge 5-1 away earlier on Tuesday. Leipzig has no points.
Elsewhere, Liverpool’s Mohamed Salah netted a double, including the winner, as it secured a 3-2 victory over Atletico Madrid, whose forward Antoine Griezmann also scored twice but was sent off in a gripping clash.
Salah fired home the winner from the spot after Griezmann had seen red for a dangerous high boot, but the action was not over in a compelling encounter, with Atletico awarded a penalty of its own, before it was overturned following a VAR review.
The win kept Liverpool top in Group B on nine points, five ahead of second-placed Atletico and Porto in third, with AC Milan pointless at the foot of the table.
Salah opened the scoring with a deflected effort in the eighth minute and Naby Keita added a second with a thunderous volley five minutes later as Liverpool looked set to crush Diego Simeone’s side.
However, Liverpool’s defence imploded shortly afterwards and Atletico came roaring back, with Griezmann at the centre of the action.
Meanwhile, Vinicius Junior scored two goals, including a brilliant solo effort, as Real Madrid thrashed Shakhtar Donetsk 5-0.
An own goal by Sergiy Kryvtsov, who turned in a Lucas Vazquez cross, gave Real a one-goal lead at the break.
Vinicius scored twice in five minutes after the restart, sweeping in from close range before beating three players to add a brilliant third.
Rodrygo swept in Vinicius’ pass before Karim Benzema slid home late on.
It was a superb second-half display by Real and in particular Vinicius, who continued his fine run of form this season.
RESULTS: Shakhtar Donetsk 0 lost to Real Madrid 5 (Kryvtsov 37-og, Vinicius Junior 51, 56; Rodrygo 65, Benzema 90+1); Atletico Madrid 2 (Griezmann 20, 34) lost to Liverpool 3 (Salah 8, 78-pen; Keita 13); FC Porto 1 (Diaz 65) bt AC Milan 0; Paris St. Germain 3 (Mbappe 9, Messi 67, 74-pen) bt RB Leipzig 2 (Silva 28, Mukiele 57); Inter Milan 3 (Dzeko 34, Vidal 58, de Vrij 67) bt Sheriff Tiraspol 1 (Thill 52); Ajax 4 (Reus 11-og, Blind 25, dos Santos 57, Haller 72) bt Borussia Dortmund 0