Sterling, who wasted chances in the 3-0 defeat by Manchester United on Sunday, looked a sharp-eyed goal-getter against second-tier opponents as he lifted the pressure on his manager Brendan Rodgers.
It was an even more comfortable evening for Tottenham Hotspur, who will face third-tier Sheffield United in a two-legged clash in January.
Nabil Bentaleb, Nacer Chadli, Harry Kane and Roberto Soldado were on target for Spurs against Newcastle who conceded four times for the second match in succession after losing 4-1 to Arsenal in north London on Saturday.
At Bournemouth, the hosts' hopes of an upset were scuppered by their own profligacy against eight-times winners Liverpool.
A glaring miss from Callum Wilson was punished on 20 minutes when Sterling headed the visitors ahead from close range.
Liverpool, who had won only two of 10 matches since they last played in the League Cup, seemed to have rediscovered their scoring touch when Lazar Markovic struck with his first goal for the club.
Sterling finished precisely six minutes after the break before Dan Gosling grabbed a consolation when his shot beat keeper Brad Jones, who should have done better.
Rodgers would have been forgiven for wanting to face anyone but league leaders Chelsea in the semi-finals.
"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," he told Sky Sports.
"I thought it was a brilliant performance from us tonight, it was always going to be a difficult job, but we controlled the game."
Tottenham had lost their last two home clashes with Newcastle, but the threat of a third was eased when a blunder from visiting keeper Jak Alnwick presented Bentaleb with the chance to hook home from four metres.
Chadli doubled the lead with a measured effort from 25 metres, Kane netted with a scuffed finish and Soldado pounced on a loose ball.
Tottenham boss Mauricio Pochettino would have been relieved to avoid Premier League opponents in the last four.
"We have respect for Sheffield United -- it is never easy," he said. "We expect a tough game."
($1 = 0.6421 pounds)
(Editing by Ed Osmond)
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