Dalglish cautious after victory
Steven Gerrard's 13th-minute penalty, after Stefan Savic had fouled Daniel Agger, was enough to beat Roberto Mancini's side 1-0 eight days after the Reds had slipped to a 3-0 defeat against the Premier League leaders at the Etihad Stadium.
But if the second half is anything to go by, when late on Liverpool had six defenders on the pitch to nullify their opponents' threat, Liverpool's bid for a first final since 2006 will be tested to the limit.
"We might have the slight advantage with the goal but it does not mean to say there will be anyone jumping through hoops at our place - we will just go and be professional in the second leg," said Dalglish.
"We have to be happy but we are not presuming that the job is done."
City were missing three key players in the suspended Vincent Kompany - who is also banned for the second leg - David Silva (ankle) and Yaya Toure (international duty) and Liverpool took full advantage early on.
Andy Carroll should have put them ahead with just five minutes gone when Stewart Downing put the striker through but he could not beat Joe Hart.
Fortunately for the striker his miss did not matter as Gerrard scored from the spot for the second successive match, having done the same in the FA Cup against Oldham, to make it four goals in as many starts of an injury-troubled season.
City pressed for an equaliser after the break as Liverpool sat back and tried to counter-attack but as the game wore on they settled for their advantage and were content to block their opponents.
"There were two different performances in each half; first half we looked a little more threatening but in the second half, having got the goal advantage, we battened down the hatches and tried to get through it with no goals against us," added Dalglish.
"Not too many teams have come here and gone away with no goals against them.
"It was a fantastic effort by the players in the second half - it might not have been the prettiest performance but it did not fall short of desire.
"Once we got ourselves in front we were determined not to lose it.
"If you come here and win 1-0 you are not going to be unhappy. It is a fantastic performance by the players against a team that are a really good side."
Dalglish also dismissed rumours of Carroll's potential exit as having "absolutely no foundation whatsoever".
Reports on Wednesday suggested the £35million club-record signing's former club Newcastle were preparing a cheeky cut-price bid to take the England international back to his native North-East.
The Magpies have played down the speculation and Carroll's representatives have also insisted they have neither been told to look for, nor are seeking, a move.
Dalglish was dismissive when asked about whether the rumours could unsettle the striker, who has scored four goals this season but has yet to find his best form since his transfer to Merseyside almost 12 months ago.
"I am worried about the people that invent them," he said.
"You have absolutely no foundation whatsoever.
"I don't need to justify what is in the paper. I have stood up - and always will stand up - for the players."