Bournemouth 1 Liverpool 3: Sterling brace helps ease pressure
The 20-year-old had not scored for club or country since September, but his brace at Dean Court ended a run of 21 matches without a goal.
His double came either side of Lazar Markovic's first goal for Liverpool, which was aided by Bournemouth defender Tommy Elphick's confusion.
With Philippe Coutinho jinking into the area, Elphick thought he had heard a whistle and appealed for offside, and the Brazilian's shot was saved by Artur Boruc before Markovic converted the rebound.
Former Everton man Dan Gosling did pull one back for the Championship leaders - his fifth goal in as many appearances in this year's competition - and the midfielder went close to a sixth when he struck the woodwork soon afterwards.
Despite coming under pressure in the second half, Liverpool will be boosted by the victory and their goalscoring exploits after being criticised for wasting a host of chances in Sunday's 3-0 defeat at Manchester United.
Possibly with Saturday's league trip to Blackpool in mind, Eddie Howe made six changes to his Bournemouth side while the visitors made three alterations from their loss at Old Trafford as Kolo Toure, Lucas Leiva and Markovic all returned.
Bournemouth's top scorer Callum Wilson had a golden chance in the fourth minute, but the 12-goal striker fired wide from close range after beating Martin Skrtel and Dejan Lovren inside the area.
Adam Lallana brought a flying save out of Boruc a minute later and after a period of sustained possession Liverpool took the lead in the 20th minute.
Markovic lofted a ball to the far post for Jordan Henderson, who nodded back across goal for a stooping Sterling to head in from five yards.
Seven minutes later the impressive Markovic doubled the visitors' advantage, calmly slotting in Coutinho's rebound following Elphick's confusion.
Bournemouth created a glorious opportunity to pull one back immediately but Yann Kermorgant blazed over the crossbar after being picked out by Simon Francis eight yards out.
Sterling grabbed his second six minutes into the second period when he turned Elphick inside out and converted a low finish into the bottom left-hand corner.
Just as it appeared Liverpool would cruise to victory, Gosling pulled one back in the 57th minute, firing in low after Matt Ritchie broke free down the right.
The hosts were buoyed by that strike and came close to a second when substitute Ryan Fraser headed narrowly wide of the upright.
Sterling scuffed a chance to seal his hat-trick wide after Lallana's neat set-up, with Gosling seeing a left-footed effort strike the post and Ritchie dragging a subsequent effort wide as Bournemouth threatened a grandstand finish.
The 2012 League Cup winners managed to weather the brief storm though, and comfortably confirmed their semi-final berth to provide a much-needed boost for manager Brendan Rodgers ahead of a Premier League clash with Arsenal on Sunday.