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Curran took five for 48 from his 10 overs, with fellow left-armer Willey grabbing four for 64 as Sri Lanka made 241-9, something of a recovery after they were 21-4 in the seventh over.
Artem Dovbyk netted in stoppage time at the end of extra time as Ukraine squeezed past 10-man Sweden 2-1.
Switzerland meet Spain in the first quarterfinal on Friday (9.30 pm IST), while Belgium clash with Italy in a mouth-watering clash (Saturday, 12.30 am IST).
Raheem Sterling and Harry Kane scored second-half goals at Wembley. It was England's first knockout round tournament victory over Germany since the 1966 World Cup final and their first ever knockout round win at the Euros inside 90 minutes.
England have lost to Germany in their last three knockout meetings at major tournaments, in 1990, 1996 and 2010, with the first two coming after penalty shoot-outs in the semifinals being particularly painful.
The former skipper said England's "batting line-up is fragile".
Haider Ali and his Peshawar Zalmi teammate Umaid Asif broke COVID-19 protocols in the Pakistan Super League in Abu Dhabi.
Goretzka's strike provided Germany with the late life-line.
England delivered an efficient but hardly scintillating display to beat the Czech Republic 1-0 with an early Raheem Sterling header.
The 21-year-old said he had persuaded his teammates to copy the hairstyle if they capture their first major trophy in 55 years.
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