Manchester United stayed in fifth place, one point above West Ham, but it was another depressingly dull day at Old Trafford where United were booed off after losing 1-0 to Southampton.
West Ham United's Mark Noble admitted with rare post-match candour that they were fortunate to escape with a 3-0 defeat by Southampton at the London Stadium on Sunday - their fourth successive reverse in the Premier League.
United are unbeaten in all competitions since Solskjaer replaced Jose Mourinho in December and their improvement is down to the Norwegian's insistence on an attacking style of play, as opposed to the defensive fare served up by his predecessor.
Photos from the English Premier League matches played on Sunday
Manchester City emerged from a trough of despair to beat Tottenham Hotspur 3-1 and move to within three points of the Premier League title at Wembley on Saturday.
Klopp has led Liverpool to back-to-back Champions League finals but is adamant that their impressive recent track record does not make them the title favourites.
Chelsea opened up a six-point lead at the top of the Premier League despite being held to a 1-1 draw against top-four hopefuls Southampton in an absorbing game at Stamford Bridge on Sunday.
Manchester United spurned a chance to move into the Premier League's top four after striker Zlatan Ibrahimovic missed a penalty in a 1-1 draw against 10-man Bournemouth on Saturday.
IMAGES from Europa League matches played on Thursday.
IMAGES from Europa League matches played on Thursday.
Tottenham Hotspur's Harry Kane pounced on a piece of comical defending to secure a 1-0 win over Sunderland in the Premier League on Sunday although he was later wheeled away on a stretcher after suffering an ankle injury.
Manchester United enjoyed perhaps their finest performance.
Jubilant fans thronged central Belgrade, bringing traffic at the main Republic Square to a halt as they waved Serbian flags and hooted their car horns.
Liverpool manager Brendan Rodgers says he has no concerns about the form of captain Steven Gerrard after a series of ineffective displays prompted calls from fans and pundits that it is time to reconsider his role in the team.
Diego Costa was on target again as Chelsea overcame a defensive West Bromwich Albion 1-0 to return to the top of the Premier League with their ninth victory in a row on Sunday. The Spaniard's second-half strike sent the Blues three points clear of Arsenal, who had taken the overall lead on goals scored on Saturday after seeing off Stoke City 3-1 at the Emirates.
Romelu Lukaku extended his impressive FA Cup scoring record by netting twice in Manchester United's 2-0 win at Huddersfield Town to earn Jose Mourinho's side a quarter-final tie at home to Brighton & Hove Albion.
Images from Wednesday's action in the UEFA Champions League.
Manchester United's Zlatan Ibrahimovic proved his hunger for silverware has not diminished with age as the Swede's double earned a 3-2 win over Southampton in an absorbing League Cup final at Wembley.
Liverpool laboured to a 2-0 win over Denmark's Midtjylland on Tuesday as a goal from Diogo Jota and a Mohamed Salah penalty secured a hard-fought Champions League Group D victory at Anfield.
Ajax Amsterdam registered a 4-1 win at De Graafschap, their first title since 2014 and a record-extending 34th overall completed the double after they won the KNVB Cup earlier this month.
Manchester United's woes deepened as stunning goals by Moussa Sow and Jeremain Lens gave Fenerbahce a 2-1 Europa League win on Thursday that left Jose Mourinho's side facing a battle to reach the knockout stages.
Swansea City came from behind to end Manchester United's unbeaten start to the Premier League season with an impressive 2-1 victory at the Liberty Stadium on Sunday.
News of all that's transpired on and off the football field.
A round-up of Saturday's Euro 2020 qualifying matches.
Images from the Champions League matches played on Wednesday
Chelsea and Manchester City tightened their stranglehold on the Premier League title race on Saturday, recording victories over struggling clubs to stretch further clear of their increasingly forlorn pursuers.
Images and summary of all the World Cup 2022 qualifiers played across continents on Tuesday.
Sergio Aguero scored a hat-trick as free-scoring Manchester City climbed to the top of the Premier League with an irresistible 6-0 victory against Watford on Saturday.
Luka Modric pulled one back for Zinedine Zidane's shellshocked side with a brilliant strike from distance early in the second half before substitute Vinicius Jr reduced the deficit to one goal.
Images from the UEFA Champions League quarter-final first leg matches played on Wednesday
Real Madrid captain Sergio Ramos risks being hit with an extra game suspension which would rule him out of a potential Champions League quarter-final first leg after admitting he got booked on purpose in Wednesday's 2-1 win at Ajax Amsterdam.
News of all that's transpired on and off the football field
Santi Cazorla scored a 94th-minute penalty as sub-par Arsenal beat Southampton on Saturday with Laurent Koscielny's overhead kick having earlier cancelled out Petr Cech's own goal.
Wasteful Portugal were held to a 1-1 home draw by Serbia after their captain and top scorer Cristiano Ronaldo limped off with an apparent hamstring injury in the first half of their Euro 2020 Group B qualifier on Monday.
With Liverpool still to visit Chelsea, who trail them by nine points with two games in hand, Juergen Klopp's side may have to beat Brighton & Hove Albion on May 13 and rely on their superior goal difference to qualify for the Champions League, assuming they do not secure it by winning this year's trophy.
A summary of Wednesday's matches in the UEFA Champions League.
Manchester City went 11 points clear with a record 14th consecutive Premier League win in an entertaining derby against Manchester United at Old Trafford.
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Liverpool substitute Adam Lallana struck deep into stoppage time to snatch a 5-4 win over shell-shocked Norwich City in an extraordinary Premier League game at Carrow Road on Saturday.
News of all that's transpired on and off the football field