After being dumped out of the UEFA Champions League by Basel last week and falling to a 3-0 defeat at fierce rivals Manchester United on Sunday, Liverpool were in desperate need of a lift.
They were provided one at Dean Court on Wednesday as Raheem Sterling's brace and Lazar Markovic's first goal for the club earned a place in the semi-finals, with Dan Gosling's goal for the Championship leaders proving irrelevant.
After a difficult nine days, Rodgers waxed lyrical about the display put on by his players on the south coast, especially that of Sterling, who was particularly wasteful in the loss at Old Trafford.
"I thought it was a brilliant performance from us tonight," he told Sky Sports.
"It was always going to be a difficult job, but we controlled the game and pressed at the right times and created chances. We thoroughly deserved it.
"Raheem Sterling has got some stick recently which I find extraordinary. He is a kid of 20 and he has been outstanding for us.
"He may not have scored the goals that some people expect, but I expect him to score goals and provide assists over the course of the season.
"His best position is on the left but he can play across the line and has the qualities to play in a central role, as he has for our youth teams."
Liverpool were drawn to face current Premier League leaders Chelsea in the last four, with Tottenham facing Sheffield United in the other semi-final, but Rodgers is not daunted about the prospect of taking on Jose Mourinho's men.
"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 said.
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