Both sides had vociferous penalty appeals turned down and Chelsea goalkeeper Thibaut Courtois showed his worth with a series of superb saves to keep the tie level heading into next Tuesday's return leg.
For record eight-times League Cup winners Liverpool, however, it was a chance missed, having dominated the clash against a strangely subdued Chelsea side who will now have home advantage and be strong favourites to reach Wembley.
The winners will face Tottenham Hotspur or third tier Sheffield United, who meet on Wednesday, in the final.
"I'm delighted with the performance, disappointed with the result," was Liverpool manager Brendan Rodgers' assessment of an evening of near-misses for the hosts.
Minutes after home captain Steven Gerrard had seen a stunning 30-metre effort tipped over by Courtois, Liverpool were playing catch-up.
Hazard calmly slotted home from the spot in the 18th minute after Liverpool's makeshift defender Emre Can clumsily bundled into the Belgian in the area.
LOUD APPEALS
The hosts had their own loud appeals for a penalty turned down when the ball hit prone Chelsea striker Diego Costa's arm at the end of the first half, but a baying Kop had to wait until the 59th minute for something to celebrate.
Sterling, refreshed from a winter break in Jamaica, received the ball inside the Chelsea half and then accelerated away from two players before knocking the ball into space and firing left-footed past Courtois.
The goal gave Liverpool their belief back and prompted a spell of frenzied pressure with Gerrard hitting the post and substitute Adam Lallana drawing a stunning one-handed save from man-of-the-match Courtois.
Chelsea coach Jose Mourinho, in typically bullish fashion, described it as a fair result, before adding that "Liverpool played well but we were in control".
The statistics told a different story, with Liverpool having six shots on target to Chelsea's one and the Merseysiders hogging 62 percent of possession, but there is little doubt Chelsea are favourites to win the tie at Stamford Bridge.
(Reporting by Toby Davis; editing by Martyn Herman and Ken Ferris)
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