Captain Gerrard, who has just returned from a long layoff with a groin injury, clinically converted a 13th-minute penalty to put seven-times winners 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 first half we didn't play well because after six months it can happen for the first time," City manager Roberto Mancini told the BBC.
"It is clear that without three or four players it's more difficult but in the end we didn't deserve to lose the game. We played poorly in the first half but the result is not right. The right result was a draw."
The winners of the semi-final will face either Crystal Palace or Cardiff City, both of the Championship, in the Wembley final. Palace take a 1-0 lead to Wales for their second leg.
City, with captain Vincent Kompany suspended after his red card in the 3-2 home defeat by United, powerhouse Yaya Toure on international duty and playmaker David Silva injured, never got going as Liverpool dominated early on.
Goalkeeper 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 when Gerrard drilled his shot low into the corner after Stefan Savic's clumsy tackle on Daniel Agger in the penalty area.
Mancini was clearly unhappy with what he was witnessing and he substituted a disgruntled Mario Balotelli before the break, the Italian striker with his trademark mohican strutting down the tunnel after being replaced by Samir Nasri.
The home side, who beat Liverpool 3-0 in the Premier League last week after a surprise defeat by Sunderland on New Year's Day, enjoyed long spells of possession after the break but rarely opened up a massed Liverpool defence.
Sergio Aguero had a chance to equalise when he latched on to a careless back-pass by Martin Kelly but was kept out by the 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 now favourites to reach the final of the League Cup for an 11th time.
"We wanted to come here and get a clean sheet and a goal which we did but we know there is no way this tie is over," Gerrard said. "We will have a very tough match in two weeks' time so the job is half done."
"It's games like this I've missed," England midfielder Gerrard, who has suffered a frustrating 12 months largely on the sidelines, added.
