Cesc Fabregas pulled one back for the Catalans but they still trailed 2-1 a minute before the end of normal time.
The former Arsenal midfielder then scored again, before David Villa netted four minutes into injury time to clinch three points and retain Barcelona’s 100% record.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.