The veteran forward took advantave of Mario Mandzukic's cold fire home a first-half treble, while Juventus bounced back from their defeat to Inter with a comfortable 4-0 win over Nordsjaelland.
Champions League round-up: Pizarro claims hat-trick as Bayern Munich blast six past Lille
Bayern Munich brushed aside pointless Lille 6-1 at the Allianz Arena to open up Group F, with Claudio Pizarro leading the rout with a first-half hat-trick.
Bastian Schweinsteiger had set Bayern on their way with a sixth minute opener and further goals were added by Arjen Robben and Toni Kroos, while Salomon Kalou replied for the French side.
The veteran Pizarro grabbed his starting spot after Bayern's top scorer Mario Mandzukic was ruled out with a cold, and underlined the credentials that have made him the Bundesliga's most prolific foreigner of all time.
"It has been some time that I have scored three goals in one half," the 34-year-old said afterwards.
"I am very happy to have scored.
"We played very well from the start and the early goal was important.
"Then the goals started coming. It was just fun to play out there tonight. I hope it continues like that."
Valencia joined Bayern on nine points with a 4-2 home win over BATE Borisov. Two goals from Sofiane Feghouli and one each for Jonas and Roberto Soldado saw off the Ukrainians, for whom Renan Bressnan and Dmitri Mozolevski got on the scoresheet.
Juventus kept the destiny of Chelsea's Group E group wide open with a comfortable 4-0 win over Norwegian side Nordsjaelland in Turin.
Juve, who now sit a point behind the top two, were never troubled with Claudio Marchisio opening the scoring after six minutes and further goals added by Arturo Vidal, Sebastian Giovinco and Fabio Quagliarella.
"We played well both technically and mentally," said Marchisio afterwards.
"It was important to win and we did well. After the defeat against Inter there was a lot of talking but people forgot that we hadn't lost for a year."
Galatasaray's 3-1 win over Cluj in Romania in Group guaranteed Manchester United's progression to the knockout stage as group winners.
There was even better news to come for Celtic following their win over Barcelona, as Benfica beat Spartak Moscow 2-0 in Portugal to leave the Scottish champions three points clear in second place.
Oscar Cardozo's second-half double was enough to provide Benfica with their first group stage win, while Spartak's slim hopes ended after having Nicolas Pareja sent off in the 76th minute.
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