Lazar Markovic scored between Sterling's first goals since September as Liverpool won on the south coast. Progress in English football's second-tier knockout competition followed a demoralizing week that saw Liverpool exit the Champions League and slip further down the Premier League.
Nabil Bentaleb, Nacer Chadli, Harry Kane and Roberto Soldado scored in Tottenham's demolition of Premier League rival Newcastle. In the draw for the two-legged semifinal in January, Tottenham was paired with third-tier side Sheffield United, which upset Southampton to advance on Tuesday.
Liverpool, which won the competition in 2012, recovered quickly from its loss at Manchester United on Sunday against the League Championship leaders.
"It's been difficult the last few weeks but I think the boys did really well today," Sterling said.
The England winger ended a 22-match goal drought for club and country in the 20th minute, heading into the net after Jordan Henderson nodded the ball across the goal.
Liverpool doubled its lead inside seven minutes when Markovic scored his first goal since joining from Benfica in July after Philippe Coutinho's low shot was blocked by goalkeeper Artur Boruc.
Sterling's second came in the 51st after he was released by Adam Lallana and twisted past Bournemouth captain Tommy Elphick before tapping into the net.
"We knew we would have to show quality tonight and the little man (Sterling) was brilliant," Lallana said. "Week in, week out he's worked hard and eventually his goals were going to come and it's a credit to how mature he is at such a young age."
Liverpool goalkeeper Brad Jones, who retained his place despite conceding three at Old Trafford, had a quiet night until being beaten by former Everton player Dan Gosling in the 57th minute.
At White Hart Lane, Tottenham reached the League Cup semifinals for the first time in five years.
A blunder by Jak Alnwick, playing only his third game in goal for Newcastle, gifted Tottenham the opener in the 18th minute. The 21-year-old Alnwick dropped a corner and Bentaleb hooked the ball past the goalkeeper.
Tottenham raced further ahead inside a minute of the second half, with Chadli powerfully striking past Alnwick.
The 21-year-old Kane netted his 12th goal of his breakthrough season in the 65th after being set up by Andros Townsend.
Kane's replacement, Soldado, completed the rout in the 70th, a minute after coming off the bench. Alnwick parried Christian Eriksen's shot and Soldado tapped the ball over the line from close range for only his fourth goal of the season.