Barcelona ended Madrid's unbeaten run in La Liga at 42 games by dealing Carlo Ancelotti's side their first loss since Septemer 23 last year.
Lewandowski scored from a superb pass by Barcelona's latest youth talent Marc Casadó in the 54th minute and headed in a second goal three minutes later to silence Santiago Bernabeu Stadium on Saturday night.
Those were goals 16th and 17th overall for the Poland striker this season, taking his league-best tally to 14. He also hit the post.
Yamal made it a rout in the 77th when the teenage star rifled a shot from an angle over goalkeeper Andriy Lunin. Raphinha got involved in the burgundy-and-blue party when he broke free with Madrid pressing forward and chipped Lunin in the 84th.
Mbappé had a frustrating first game against Barcelona in a white shirt.
The France star was repeatedly caught offside by Barcelona's synchronised back four and failed to beat second-choice goalkeeper Iñaki Peña one-on-one with his team down two goals.
Kylian Mbappe endured a disappointing first clasico experience. (AP PHOTO)
Hansi Flick, meanwhile, had a night to remember in the first clasico for the German coach.
Barcelona still has financial troubles that cast doubt over its future but its team have made huge steps forward since Flick replaced Barcelona great Xavi Hernández in the summer.
Barcelona opened a six-point lead at the top of La Liga over second-placed Madrid after 11 rounds.
"We had a tough week with tough games, but we knew how to pull it out. We played a great game," Casadó, an unknown reserve team player until Flick picked him to be his holding midfielder, said after his impressive outing.
Casadó was one of five homegrown players that Flick put in his starting lineup, the same 11 that beat Bayern Munich 4-1 in the Champions League three days before.
A total of six players under 22 years old started for Barcelona, led by 17-year-olds Yamal and Pau Cubarsí, and showed no signs of being intimidated by the scenario.
Yamal upstaged Mbappé this summer when Spain beat France en route to winning the European Championship, and he did it again when he displayed the finishing touch that Mbappé was lacking.
Mbappé himself was ruled offside no less than five times in the first half hour, as Barcelona pressed high up the field. Vinícius Júnior, who featured in a fascinating duel with Cubarsí, missed a golden chance in the 21st when he shot past the post from close.
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