Villas-Boas has found his position under increasing scrutiny in recent weeks following a poor run that included three defeats in four Premier League matches.

Liverpool’s second win at Chelsea in the last nine days will inevitably bring renewed speculation about the young Portuguese coach’s future.

Elsewhere, Manchester City moved into the semi-finals with a 1-0 win over Arsenal, Sergio Aguero hitting a late winner for the Premier League leaders.

Liverpool boss Kenny Dalglish had threatened to field a weakened team in response to the fixture scheduling that forced his team to travel to west London just 48 hours after a draw with Manchester City. But in the end he sent out a strong line-up including Jamie Carragher, Andy Carroll and Craig Bellamy.

Chelsea had an early penalty appeal turned down when Sebastian Coates appeared to foul David Luiz, but referee Phil Dowd did point to the spot in the 20th minute when Blues defender Alex handled a Jose Enrique cross. However Carroll’s troubled spell with Liverpool took a turn for the worse as his penalty was saved by Ross Turnbull.

Florent Malouda went close for Chelsea when his second-half effort hit the bar, but Liverpool took the lead through Maxi Rodriguez in the 58th minute.

Rodriguez also scored in Liverpool’s 2-1 win at Stamford Bridge in the Premier League nine days ago and the Argentina midfielder couldn’t miss this time after Bellamy set him up for a simple tap-in.

Chelsea striker Fernando Torres drew a blank against his old club and it was Liverpool who scored again through Martin Kelly, the young defender heading in his first senior goal from Bellamy’s free-kick in the 63rd minute.

At the Emirates Stadium, City manager Roberto Mancini made 10 changes after bemoaning the fixture schedule but it was hardly an under-strength team since it included the likes of former Arsenal stars Samir Nasri and Kolo Toure, as well as Edin Dzeko and Owen Hargreaves.

Inevitably Nasri was jeered on his first return to the Emirates since his acrimonious move to City in August, but he had the last laugh as City went through.

City’s understudies dominated possession and they looked even more formidable when Mancini sent on Argentina forward Aguero before half-time.

And it was Aguero who got the decisive goal in the 83rd minute as his excellent strike completed a sweeping move involving Dzeko and Adam Johnson.

Championship side Cardiff enjoyed a 2-0 win over struggling Blackburn that left under-fire Rovers boss Steve Kean facing a fight to avoid the sack.

With Rovers sitting bottom of the Premier League with just one win this season and Blackburn fans calling for his dismissal, Kean could ill afford an embarrassing exit at the Cardiff City Stadium even though the club’s Indian owners have remained publicly supportive.

So Kean must have been shifting uncomfortably in his seat when Scotland striker Kenny Miller put Cardiff ahead in the 19th minute with a cool finish after Morten Gamst Pedersen squandered possession in midfield.

Miller nearly got a second when he went around Blackburn goalkeeper Mark Bunn, only for Rovers to twice clear his efforts off the line.

Kean sent his players out early before the start of the second half.

But Anthony Gerrard piled on more misery for the Rovers boss in the 50th minute when he headed home to clinch Cardiff’s first win over a top-flight club in the League Cup since 1986.

Manchester United host Championship club Crystal Palace in the last quarter-final tie on Wednesday.