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.
"It's normal, it's about respect,'' Mbappe said about allowing Messi to take the first penalty. "He's the best player in the world. I've always said it's a privilege to play with him.
"There is a penalty, he took it and scored. The second, he wanted to give it to me, and I missed."
Mbappe added that Leipzig made it hard for PSG to earn the comeback win.
"[Were we] lucky? No, you make your own luck," he said. "But of course, it was difficult, Leipzig played well. We had some low spells which were perhaps a little bit too low.''
"They pressed really hard and targeted Marco Verratti," said PSG coach Mauricio Pochettino, who had to deal with the absence of the injured Neymar and suspended Angel Di Maria.
"It was a very tough game but the team showed their character again."
Leipzig's American coach Jesse Marsch added: "There's a massive difference between us and PSG but the fact that we almost made it says a lot. We need to trust this young squad."
Before Mbappe and Messi linked up like two old teammates, PSG often looked out of sorts, incapable of dealing with Leipzig's high pressure game.
Marsch's side pressed hard from the outset and after eight minutes, PSG's Keylor Navas had already made three saves.
It was a double-edged sword for the visitors, however, as PSG threatened on the break and they opened the scoring after nine minutes.
Leipzig did not back down and in the 27th minute Silva's volley crashed against Navas's left-hand post. A minute later Silva turned Angelino's cross in at the far post to give the German side a deserved equaliser.
Leipzig were the better team and Christopher Nkunku came close to giving them the advantage three minutes before the break when his low shot went just wide after a series of dribbles.
PSG started the second half well, with Messi finding space between the lines, but it was Leipzig who took the lead in the 57th as Mukiele latched onto another Angelino cross at the far post.
The hosts continued to struggle but a Tyler Adams back pass was intercepted by Mbappe who sped into the area and fed Messi, with the Argentine tapping the ball into the empty net after Gulacsi parried his shot onto the inside of the post after 67 minutes.
It was Mbappe who had set up Messi for PSG's second goal in their 2-0 victory against Manchester City in the previous round of matches.
Seven minutes later, Mbappe was brought down by Mohamed Simakan and Messi converted the resulting penalty in clinical fashion before again pointing to his strike partner, who has set up his three PSG goals so far.
PSG were awarded another penalty four minutes into added time after a VAR review for a Josko Gvardiol foul on Achraf Hakimi but Mbappe skied it over the bar, although that did not tarnish a sublime overall performance from the 22-year-old.