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.

"Their goal was double offside, but in the end, we had luck in the penalty shootout," Klopp told Sky Sports.

"Over the whole 120 minutes, the players, crowd and Liverpool deserved it. Wembley is a cool place to play football, but we go there to win. It's not much fun to lose."

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 fell short in their bid to reach their first League Cup final since they won the tournament in 1972.

"I'm really proud of what the players produced," said Hughes. "I can't fault any of them. I thought we were the dominant team, but we're not the ones going to Wembley, unfortunately."

It took until the first minute of extra time for Liverpool to register a shot on target from open play, Stoke goalkeeper Jack Butland saving a toe-poke from Roberto Firmino.

But Stoke substitute Marco van Ginkel twice went closer, and each time from Peter Crouch flick-ons, first scuffing a shot against the right-hand post and then testing Mignolet with a volley.
                  
Sakho block 
                  
With Liverpool substitute Jordon Ibe skewing a shot over late on, the game went to penalties and after each side had squandered a spot-kick -- Mignolet saving from Crouch, Emre Can hitting the post -- it went to sudden death.

Van Ginkel and Lucas Leiva scored, but Belgium international Mignolet -- maligned for some shaky recent performances -- plunged to his left to parry Muniesa's spot-kick and Allen stepped up to settle the semi-final by planting the ball into the top-right corner.

Liverpool's fragility beneath the high ball was demonstrated anew in Saturday's madcap 5-4 win at Norwich City and Hughes sought to target their aerial vulnerability by awarding one-time Anfield native Crouch only his fifth start of the season.

The gangly former England striker got his head to plenty of balls, but it was from an approach more in keeping with Stoke's football this season that the hosts procured their best first-half chances.

Two opportunities fell to Jon Walters. From Arnautovic's through-ball in the 14th minute the Irishman saw a shot deflected over and when Erik Pieters's pass was allowed to bounce through to him eight minutes later, he shot wide under pressure from the sliding Mamadou Sakho.

A tame first half was trundling towards its conclusion when Stoke struck in stoppage time, Bojan Krkic crossing from the right and Arnautovic stealing in behind Kolo Toure to side-foot home.

Liverpool's defenders appealed in vain to the linesman, but television replays confirmed that the Austrian had been offside.

The home fans' boos became roars early in the second half as Liverpool looked to reassert their advantage, with Butland diverting a drilled shot by Firmino onto the post.

The hosts' defensive weaknesses remained apparent, however, and it took a last-ditch block from Sakho to thwart Walters after Liverpool had failed to clear a free-kick from Pieters.

With a late Liverpool penalty appeal for handball against Pieters dismissed by referee Jon Moss, the game went to extra time.