Arsenal beat Tottenham Hotspur to keep alive their own slender hopes of retaining their Premier League title.
In Tuesday's other match, Middlesbrough drew 1-1 with Fulham.
With 10 games left, United are now 12 points behind league leaders Arsenal and three behind second-placed Chelsea.
Striker Georgios Samaras scored his fourth goal in eight games for his new club to give them a 2-1 victory over Aston Villa.
2004 Champions League finalists Monaco were eliminated 2-1 on aggregate by Swiss side Basel.
Manchester United beat Bolton Wanderers 2-1 to go three points clear in the premier league.
The holders and runners-up made their exit from the League Cup in the fourth round after being beaten by Bolton Wanderers and West Bromwich Albion respectively.
Champions Chelsea and second-placed Manchester United beat Middlesbrough and Portsmouth respectively in the Premier League.
A review of the weekend action in the leading European leagues.
Michael Owen scored a Liverpool beat Leeds United 3-1 to end their three-match losing streak in the Premier league.\n\n
Bournemouth's remarkable rise from the brink of obscurity was virtually completed when they effectively sealed promotion to the Premier League for the first time in their 116-year history on Monday.
Brief reports of English Premier League matches played over the weekend\n
Minor league Eastleigh suffered FA Cup heartbreak when they conceded a late equaliser to four-times winners Bolton Wanderers on a gluepot pitch at their snug Ten Acres ground on Saturday.
They were held 2-2 by Bolton Wanderers in the English Premiership.
Arsenal have the chance to return to the top when they visit Bolton Wanderers on Saturday.
Bolton Wanderers captain's signing is subject to being granted a new work permit.
His reported wages of 65,000 pounds ($115,300) a week make Viduka a soft target for those Leeds fans shocked by the incredible decline of the club.
The former AC Milan and Barcelona forward has returned home after last week's talks with the premier league club.
Tottenham Hotspur cruised to a comfortable 5-1 victory over Aston Villa at White Hart Lane on Sunday
The Liverpool and England midfielder was penalised for his two-footed tackle on Everton's Gary Naysmith.
The Swedish striker gave Arsenal a 1-0 win over Bolton Wanderers in the FA Cup quarter-finals.
Raheem Sterling and Philippe Coutinho helped Liverpool reach the FA Cup fifth round.
But title rivals Chelsea and Manchester United were held in Saturday's Premiership matches.
"I've had talks with Hierro and his representatives over the weekend and he is keen to come to Bolton," Bolton manager Sam Allardyce said.
"It's not 100 percent finished but looks very, very doubtful from our point of view," said Bolton Wanderers manager Sam Allardyce.
A remarkable lobbed goal by Henrik Pedersen after only four minutes has given Bolton Wanderers a 1-0 home win over Newcastle United in the premier league.
Middlesbrough are through to the League Cup final with a tense 2-1 victory in their semi-final, second leg match.