The FA Cup final at Wembley Stadium on Saturday was the last chance for the Reds to win a quadruple.
Cesar Azpilicueta missed and Mason Mount had his spot kick saved by the Greeks.
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The sides played out an entertaining stalemate over 120 minutes before being decided on penalties again.
The early stages of the game were dominated by Luis Diaz. A through-ball from Trent Alexander-Arnold put Diaz through on goal in the eighth minute, but Mendy denied him before Naby Keita put the rebound wide.
After excellent play between Mount and James on the right wing, Christian Pulisic went close for the Blues on 23 minutes.
Alisson was off his line quickly to deny the Chelsea defender after Marcos Alonso was given a sight of goal.
On 33 minutes, a blow was dealt to the club when Mohamed Salah was forced off with an injury. He was replaced by Diogo Jota and he put a good chance over just before the half-time break.
Both sides toiled and could only create half-chances for much of the second period after the break.
Andy Robertson struck the post from close-range when substitute James Milner's cross found him.
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Spot kicks were used for the second time this season after neither side was able to find a breakthrough in the additional 30 minutes.