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New signings Hugo Ekitike and Jeremie Frimpong scored for the Premier League champions, but Palace responded through Jean-Philippe Mateta and Ismaila Sarr before winning an error-strewn shootout 3-2.
Mohamed Salah blazed over from the spot, while Alexis Mac Allister and Harvey Elliott were denied by an inspired Dean Henderson as Palace built on winning their first ever major trophy by beating Manchester City in May's FA Cup final.
The traditional curtain-raiser to the English football season was given extra significance after a summer marked by tragedy for Liverpool. Forward Diogo Jota was killed in a car accident alongside his brother Andre Silva.
Reds legend Ian Rush and Palace chairman Steve Parish laid wreaths on the side of the pitch before kick-off, while the Liverpool end was awash with banners and flags paying tribute to the Portuguese international.
