Raheem Sterling eased the pressure on under-fire manager Brendan Rodgers as Liverpool swept into the League Cup semi-finals with a 3-1 win over Championship leaders Bournemouth on Wednesday.

Rodgers has endured stinging criticism in recent weeks after Liverpool dropped down the Premier League table and crashed out of the Champions League in the group stages.

A dismal 3-0 defeat at arch-rivals Manchester United on Sunday prompted Rodgers to admit Liverpool’s squad lacked the unity that almost carried them to the English title last season.

In the circumstances, an embarrassing defeat to second tier Bournemouth would have been a major blow to the beleaguered Rodgers’ hopes of holding on to his job, but he was able to leave Dean Court with his head held high as Sterling’s double sandwiched a Lazar Markovic strike in a one-sided quarter-final.

“I thought it was a brilliant performance. It was always going to be a difficult job, but we controlled the game. We thoroughly deserved it,” said Rodgers, whose side will face Chelsea in the two-legged semi-finals in January.

“Raheem Sterling has got some stick recently which I find extraordinary. He is a kid of 20 and he has been outstanding for us.

“It didn’t matter who we got in the draw, it was just important to get there – to win this competition you have to win games against big sides.”

Despite their recent struggles, Liverpool had won at Dean Court in the FA Cup last season and, after Bournemouth’s Callum Wilson missed a good chance early on, the visitors took control.

England forward Sterling opened the scoring with a glancing header from close range after Jordan Henderson nodded Markovic’s cross back across goal in the 20th minute.

Rodgers’ relief was clear to see as he celebrated Sterling’s first goal since September by vigorously punching the air on the touchline.

And, with the Reds cutting through the Bournemouth defence at will, Rodgers was on his feet again seven minutes later to salute Liverpool’s second goal.

Philippe Coutinho ignored Bournemouth’s appeals for offside as he cut inside to test Artur Boruc with a low shot and when the ball spun out to Markovic, the Serbia winger kept his cool to sidefoot home his first goal for Liverpool from the edge of the penalty area.

With Liverpool’s supporters making their support for Rodgers loud and clear, Sterling ran clear to net their side’s third goal in the 51st minute as he glided past Tommy Elphick before slotting home.

Former Everton midfielder Dan Gosling got one back for Bournemouth in the 57th minute when his tame shot trickled past a weak attempted save from Brad Jones.

In Wednesday’s other quarter-final, Tottenham thrashed Newcastle 4-0 at White Hart Lane as the Magpies suffered more north London misery.

Five days after a 4-1 beating at Arsenal, Alan Pardew’s side were on course for another heavy defeat from the 18th minute.

Jak Alnwick, a 21-year-old making just his third appearance in goal for Newcastle, made a hash of catching a corner and allowed the ball to slip out of his grasp to midfielder Nabil Bentaleb, who hooked home from four yards for his first Tottenham goal.

Newcastle conceded again 36 seconds into the second half when Nacer Chadli drove a perfectly-placed shot into the corner of Alnwick’s net.

Harry Kane put the result beyond doubt when the Tottenham striker scuffed his shot past Alnwick in the 65th minute and Spanish forward Roberto Soldado added the fourth five minutes later.

Tottenham’s reward was a semi-final tie against third tier Sheffield United, who enjoyed a shock 1-0 win over Southampton on Tuesday.

“We have respect for Sheffield United. They deserve to be in the semi-finals. We expect a tough game,” Tottenham manager Mauricio Pochettino said.

Also on Tuesday, Premier League leaders Chelsea had advanced to the last four with a 3-1 win at Championship side Derby.