Joe Allen scored the decisive spot-kick after Marc Muniesa had his penalty saved by Simon Mignolet.
After Marko Arnautovic had cancelled out Liverpool’s 1-0 aggregate lead with a goal that seemed to have been scored from an offside position, the game went to extra time and then penalties, and with Mignolet thwarting Muniesa, it was left to Joe Allen to net the decisive spot-kick.
Following some slapdash recent displays, it was a much-needed breakthrough for Liverpool manager Jurgen Klopp, appointed in October, whose side will face either Manchester City or Merseyside rivals Everton in the final on February 28.
It will be Liverpool’s first major final since 2012, when they won the last of their eight League Cups and lost to Chelsea in the FA Cup final.
Mark Hughes’s Stoke had eliminated holders Chelsea in round four, but despite recording a first win inside 90 minutes at Anfield since March 1959, they were unable to secure a place in what would have been their first League Cup final since they won the tournament in 1972.