But they went ahead via a controversial penalty when Micah Richards blocked Daniel Agger's shot with his arm after the ball ricocheted off his foot.
Steven Gerrard tucked that away and, although Edin Dzeko levelled the tie again from close range, Craig Bellamy scored against his former team with 16 minutes left to book their meeting with another of his old clubs, Cardiff, on 26 February.
Afterwards City boss Roberto Mancini claimed it was "impossible" to award Liverpool the spot-kick, but Dalglish said his side deserved to be in their first League Cup final since 2005.
"I think it was a great spectacle for everybody and I think overall we deserved to win," the Scot told BBC Sport.
"[Manchester City goalkeeper] Joe Hart put in a fantastic performance which wasn't pleasing for us but must have been very pleasing for them.
"The attitude and commitment we showed when going behind both times and coming back to get on level terms was fantastic," added Dalglish, part of the Liverpool team which won the first four of their record seven League Cup victories in successive years between 1981 and 1984.
Liverpool skipper Gerrard, twice a League Cup winner himself (in 2001 and 2003), believed it was his team's commitment to attack which won them the tie and said that it had been "too long" since Liverpool's last trip to Wembley.
Wembley final a dream come true - Gerrard
"It was a tough night," he commented. "Manchester City are a fantastic team with very dangerous players, but I thought we were superb tonight and stuck together. The best form of defence was attack, that's what we've done and we got our rewards."
Having scored a memorable late equaliser in Liverpool's last major final in 2006 when they beat West Ham on penalties after a 3-3 draw at the Millennium Stadium, Gerrard said it would be a dream to finally lead his team out at England's national stadium.
"We're delighted to go to Wembley," he added. "It's been too long for a major cup final for this club so I'm really happy for our supporters.
"We had a lot of cup finals at Cardiff, but it's extra special going to Wembley.
"I've been dreaming of doing it for Liverpool and now that it's here I'm really looking forward to it, but there are a lot of big games before that."
Liverpool face the first of those on Saturday when they host Manchester United in the fourth round of the FA Cup.
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