The French side led through an early Hakon Haraldsson goal in the Italian capital but Roma had the chance to equalise late on after a handball by Aissa Mandi.
Liverpool maintained their perfect Champions League record to guarantee a top-two finish in the league phase and automatic last-16 qualification with a 2-1 win over Lille at Anfield on Tuesday.
Manchester United are set to sign Lille's promising teenage defender Leny Yoro in a £52 million ($69 million) swoop, according to reports on Wednesday.
The visitors were punished when Mbappe turned in Juan Bernat's cross to make it 3-3 before Messi's free-kick five minutes into stoppage time sealed a dramatic victory for PSG.
