Manchester City defeated Salford City in the FA Cup, while Burnley suffered a surprising defeat to Mansfield Town. West Ham United and Norwich City also secured victories in their respective matches.
City, Liverpool all cruise in respective FA Cup matches played on Saturday.
Premier League clubs West Bromwich Albion and Watford were knocked out in the League Cup second round by lower-league opponents on Tuesday as a strong Liverpool side crushed championship Burton Albion 5-0.
Marcus Rashford scored twice as holders Manchester United brushed aside second-tier Burton Albion 4-1 at Old Trafford to reach the fourth round of the League Cup on Wednesday.
League One Burton, managed by Nigel Clough, the son of former Nottingham Forest boss Brian, must endure another 90 minutes at their own ground on Jan. 23 before City formally book their place in the final.
Holders Manchester City held their nerve to beat Leicester City 3-1 on penalties and reach the League Cup semi-finals after an entertaining last-eight tie ended 1-1 on Tuesday.
Spanish second division side Rayo Vallecano refused to turn up for the first day of group training due to a dispute with the club hierarchy over a pay cut for staff and players, a club source said on Monday.
Manchester United crushed minor league Burton Albion 5-0 in their FA Cup third-round replay
Manchester United will travel to third-tier MK Dons in the Capital One Cup second round following the draw on Wednesday, the first time they have been involved at this stage of the competition for 19 years.
Liverpool's Daniel Sturridge has urged manager Juergen Klopp to deploy him in his favoured centre forward position, where he operates on "autopilot".
Such is the impact Sadio Mane has made on Liverpool that the 24-year-old is an automatic pick to face Tottenham Hotspur in the Premier League on Saturday while experienced international teammates scrap it out to play alongside him.
Liverpool needed two extra-time goals to beat NottsCounty 4-2 at Anfield on an eventful night for the Premier League clubs in the League Cup second round on Tuesday.
Michael Carrick and Ander Herrera have hardly featured in Manchester United's early Premier League campaign but manager Jose Mourinho has assured the duo they are vital to his plans.
The visitors' display was enough to earn them applause from their fans at the end.
Liverpool manager Juergen Klopp called for his players to "stay calm" and does not expect to expect wholesale changes against Burton Albion on Tuesday as they seek to quickly bounce back from their 2-0 defeat at Burnley.
Images from the League Cup matches played on Tuesday
Fulham forgot their Premier League struggles to sweep into the FA Cup fourth round with a 3-0 victory over fellow top-flight strugglers NorwichCity in a third-round replay on Tuesday.
Manchester United's current malaise was laid bare as they struggled past third-tier Rochdale 5-3 on penalties after a 1-1 draw in the League Cup third round.
Hazard's dribbling and movement, on and off the ball, constantly posed questions for the Liverpool defence and the winger, who was so central to Chelsea's title triumph last year, certainly looks back up to full speed.
Here are five talking points from the weekend's action:
Liverpool manager Brendan Rodgers was an immensely proud but competitive dad at Anfield on Sunday as his Premier League side overcame the FA Cup challenge of his son's third tier Oldham Athletic.
Steve Finn helped England dominate Australia in another Ashes Test at Edgbaston on Friday to take a 2-1 series lead. A day later, there were celebrations across China as Beijing won the rights to host the 2022 Winter Olympics. And, on Sunday, Premier League club Arsenal finally won bragging rights after beating Chelsea 1-0 to clinch the Community Shield to kick-start their season on a positive note. Rediff.com has all this and a lot more action from the week gone by, snapped right here!
Frank Lampard and Edin Dzeko scored twice as holders Manchester City sailed into the English League Cup fourth round with a crushing 7-0 win over Sheffield Wednesday and Chelsea beat Bolton Wanderers 2-1 on Wednesday.
Tottenham Hotspur gained a slender advantage in their League Cup semi-final against London rivals Chelsea as Harry Kane's penalty earned them a gritty 1-0 first-leg win.
Third-placed Leicester go to Hodgson's sixth-placed Crystal Palace on Sunday
Britain's world number three Andy Murray stressed the importance of gender equality in tennis, saying the work ethic required to succeed is the same for both men and women.
News of all that's transpired on and off the football fieLd