The Swede was quiet, even off-colour, until scoring twice in four minutes at the end of the first half, converting a controversial penalty and then smashing a left-footed shot from the edge of the area straight into the top corner.
Blaise Matuidi earlier stunned the BayArena just three minutes into this first leg with the opening goal, and that set the tone for the match.
PSG had no need to get out of second gear against opponents who are no strangers to the latter stages of the Champions League but froze badly this time and appeared to accept their fate.
Leverkusen were reduced to 10 men in the 59th minute, defender Emir Spahic harshly sent off for two yellow cards, and their misery was complete when PSG substitute Yohan Cabaye added a late fourth with his first goal for his new club.
The hosts got off to a terrible start as Simon Rolfes gave the ball away in his own half to Matuidi, who started a move which ended with him running onto Marco Verratti's through ball and side-footing a low finish into the left corner, with goalkeeper Bernd Leno slow in getting down.
The hosts were looking very nervous as their opponents pressed them high up the pitch and Leno had a let-off in the 11th minute after spilling a cross.
He dropped a ball in from Gregory van der Wiel straight at the feet of Ibrahimovic who blasted his shot from six yards straight at a defender.
At the other end Stefan Kiessling brought a save out of Salvatore Sirigu, who tipped his header over the top, but that was as good as it got for the home side, who lacked the invention to break down their visitors.
Leverkusen started to have more possession, but it was all in front of their opponents, who were so comfortable they were prepared to dribble out of their own area on occasions.
PSG's second goal arrived from the spot in the 39th minute, but the penalty award was by no means clear-cut.
Maxwell cut the ball into the area from the byline on the left and Ezequiel Lavezzi went down under pressure from Spahic.
There were hardly any appeals from the PSG players, apart from Lavezzi, but the referee spotted a shirt pull on the Argentinian and pointed to the spot
Ibrahimovic's penalty was perfect, fired into the bottom-left corner.
And the Swede stamped his brilliance on the match three minutes before the break. He started an attack, laying the ball out wide left to Maxwell, whose ball in was controlled and held up well by Matuidi. The midfielder laid the ball back to the edge of the area where Ibrahimovic blasted a first-time left-footed shot into the top-right corner. This time Leno had no chance.
Kiessling had penalty appeals waved away early in second half before things went from bad to worse for the Germans.
Spahic, booked in giving away the penalty, was shown a harsh second yellow card after going shoulder-to-shoulder with Lucas Moura.
The Bosnian centre-back looked to brush off the attentions of his opponent and, although his arm was high, Moura's fall to the turf was theatrical in the extreme.
PSG were then denied a fourth by an offside flag after Matuidi's shot was saved by Leno and then bounced in off Phillipp Wollscheid.
Leno did well to save with his legs from Javier Pastore, who slid in to get on the end of a low Ibrahimovic cross.
But substitute Cabaye made it four in the 88th minute, side-footing a shot into the top corner from just inside the area after Moura had laid Maxwell's cross into his path.
It left PSG all but in the last eight, with Leverkusen needing to stage the greatest second-leg fightback in Champions League history to deny them.
- Sports & Recreation
- Blaise Matuidi
- Yohan Cabaye