The Pole has now scored 10 goals in his last three games and his magic sent Pep Guardiola’s team to the top of Group F.
Goals from Douglas Costa, Mario Gotze and Lewandowski’s first two had given the German giants a 4-0 lead at half-time.
The Croatian champions, who beat Arsenal in their opening match, went into this clash on a 45-game unbeaten run stretching back to last November but that was brought to a shuddering halt at the Allianz Arena.
Barcelona averted another upset with two late goals to beat Bayer Leverkusen 2-1 in their Champions League Group E clash.
Kyriakos Papadopoulos headed the Germans into a 21st-minute lead and Barca, without talisman Lionel Messi through injury, struggled to get back into the game. But Carnicer Sergi Roberto eventually equalised in the 81st minute and 90 seconds later Luis Suarez secured three points for the off-colour Catalan giants.
In the other Group E game Belarusian champions BATE Borisov shocked Italian side Roma with a 3-2 win at the Borisov-Arena. The current runaway leaders in Belarus hit the visitors with two goals inside the first 12 minutes from Igor Stasevich and Filip Mladenovic before the latter struck a third just after the half hour.
Gervinho and Vasilis Torosidis scored for Roma, but the defeat left the Serie A outfit bottom of the group with one point from two games as BATE enjoyed their first win over Italian opposition at the 10th attempt.
Valencia got their Champions League campaign up and running with a 1-0 victory at Lyon in Group H.
A goal from Sofiane Feghouli just before half time was enough to move them into second place behind the Russians.
Lyon, meanwhile, are still without a win in this season’s competition and have just a single point from their opening two games.
Staying in Group H, Oleg Shatov’s second-half goal earned Zenit St Petersburg a 2-1 victory over Gent to strengthen their position at the top.
Zenit took the lead in the 35th minute through Artem Dzyuba’s close-range header.
Thomas Matton’s fine 56th-minute volley drew Gent level before Shatov’s cool finish 11 minutes later sealed all three points for the hosts.