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RB Leipzig and Porto traded goals in the first half, but Jean-Kevin Augustin's proved to be decisive for the Germans.
Three goals in the opening 19 minutes ended the game as a contest as Mohamed Salah, Philippe Coutinho and Roberto Firmino inspired an easy win.
Hugo Lloris and Keylor Navas each put on a show between the sticks as Real Madrid and Spurs battled to a draw.
Raheem Sterling and Gabriel Jesus got Manchester City off to a flying start as they held off a late surge from Napoli.
Bernard scored both goals for Shakhtar as they came from behind for a Champions League win at Feyenoord.
Tottenham held on for a 1-1 draw against Real Madrid to draw level on seven points with the holders at the top of the Champions League Group H table.