Steven Gerrard's penalty early in the first half was enough to put Liverpool in the driving seat for the second leg at Anfield in a fortnight.
City, who were knocked out of the FA Cup by Manchester United on Sunday, improved after the break following a woeful first half but were still well short of the form that has propelled them to the top of the Premier League.
The winners of the semi-final will face either Crystal Palace or Cardiff City, both of the Championship (second division), in the Wembley final. Palace take a 1-0 lead to Wales for the second leg of their semifinal.
City, with captain Vincent Kompany suspended after his red card in the 3-2 home defeat by United, Yaya Toure on international duty and playmaker David Silva injured, never got going as Liverpool dominated the early stages.
Keeper Joe Hart had to make two fine saves in the opening minutes, once from Gerrard and then from a deflected Stewart Downing effort, but he was beaten from the spot in the 10th minute when Gerrard found the corner after Stefan Savic's clumsy tackle on Daniel Agger in the penalty area.
City manager Roberto Mancini was clearly unhappy with what he was witnessing and he substituted a disgruntled Mario Balotelli before the break, the Italian striker strutting down the tunnel after being replaced by Samir Nasri.
The home side, who beat Liverpool 3-0 in the Premier League last week, enjoyed long spells of possession after the break.
Sergio Aguero had a chance to equalize when he latched on to a careless back pass by Martin Kelly but was kept out by alert Liverpool keeper Pepe Reina.
Micah Richards then forced a fine save from Reina with a close-range header but Liverpool held out and are favorites to reach the final of the League Cup for an 11th time.
Manchester City's Richards challenges Liverpool's Carroll during their English League Cup semi-final first leg soccer match in Manchester (Photo: Reuters)