Liverpool will take a 1-0 advantage back to Anfield after defeating Manchester City in their Carling Cup semi-final on Wednesday.
A first-half penalty from Steven Gerrard was all that separated the sides at the Etihad Stadium after City defender Stefan Savic clumsily felled a charging Daniel Agger in the box.
The loss was City's second at home in four days following the defeat to local rivals Manchester City on Sunday, while the visitors found more cup success after their 5-1 win over Oldham last Friday.
City would have been confident heading into the match having defeated Liverpool 3-0 at the same ground just a week ago.
Roberto Mancini's side had a different look for this match though, with Yaya and Kolo Toure away on African Cup of Nations duties with the Ivory Coast and Vincent Kompany suspended.
Liverpool pressed from the off and Carroll was handed a great chance to open the scoring after the striker was sent through on goal by Downing.
Carroll shrugged off a challenge from Savic, but could not beat the oncoming Joe Hart with a low right-foot shot.
City hard barely attacked in the Reds' third when Gerrard fired a stinging shot at Hart which dipped at the far corner, but the goalkeeper reacted quickly enough to palm the Liverpool skipper's effort out for a corner.
From the set-piece, a rehearsed move between Gerrard and Downing had the latter testing Hart for the third time for the match.
The winger's wicked attempt on the volley from outside the box was deflected by a Liverpool attacker, but the England international saw it just in time to instinctively push it out for another corner.
There was nothing Hart could do when Liverpool was awarded a penalty on 12 minutes, with Savic's high-footed challenge bringing down Agger as he burst into the box.
Gerrard stepped up to the spot-kick and slotted the ball home past Hart to put the Reds in the lead.
City finally tested Pepe Reina with a shot on target from substitute Samir Nasri, who came on after 39 minutes in place of Mario Balotelli, but that was the best they could muster in a frustrating opening 45 minutes.
City pressed forward as the half wore on, with Liverpool content to sit back and defend their lead.
But despite having the bulk of possession in the second 45 minutes, Mancini's side failed to create many clear-cut chances.
Aguero had the best chance late on, but his close-range effort flew just inches over the bar and left the Reds to be the much happier side going into the second leg in two weeks.