The night did not start well for last year’s runners-up when they fell behind to Lionel Messi’s 27th-minute penalty.
Messi, regarded as one of the greatest players of all time, was then upstaged on his own turf by France World Cup winner Mbappe as PSG ruthlessly exposed the hosts.
Mbappe danced past Clement Lenglet and smashed in the equaliser before stroking home from 10 yards to turn the game around.
On-loan Everton striker Moise Kean continued his resurgence by heading in unchallenged at the far post to put the Ligue 1 champions in control of the tie.
With Barca looking to pull a goal back, they were caught on the counter-attack and Mbappe curled in the goal of the night with his third.
The second leg takes place on 20 March in Paris, by which time PSG hope to have Neymar back from a thigh injury to face his former side.
Elsewhere in Budapest Arena, Liverpool ended their dire recent run with a confidence-boosting victory over RB Leipzig to give themselves a superb chance of reaching the last eight of the Champions League.
Both Sadio Mane and Mohamed Salah were on target for the reds.