Liverpool are two points ahead of Manchester City, who still have a game in hand.
Callum Hudson-Odoi, 18, has yet to make a Premier League start, but has impressed in Cup competitions and the Europa League for Chelsea.
Mauricio Pochettino's men threw away an early lead through Harry Kane to lose 2-1 at Southampton.
Chelsea remain in fourth place, but they are now just three points ahead of fifth placed Arsenal in the race to qualify for next season's Champions League.
David Silva put City ahead after 10 minutes but they passed up a number of chances to add to the lead.
Victory against Southampton saw the West Ham climb into ninth place in the Premier League table.
Southampton had twice taken the lead through Danny Ings.
Jose Mourinho's men fell 16 points behind Premier League leaders Manchester City.
Sergio Aguero, with his 150th league goal, David Silva and Leroy Sane were also on the mark as Manchester City continued their unbeaten start.
Leicester City's Premier League match away to Cardiff on Saturday will go ahead as planned.
Ross Barkley set up Eden Hazard's 30th-minute opening goal before adding the second as Chelsea beat Southampton on Sunday.
Mohamed Salah left a young Liverpool fan in tears after handing over match shirt.
Liverpool have now won their first six league games for the first time since 1990.
Liverpool are looking to register their sixth straight win in the Premier League this season.
Glenn Murray proved a thorn in Southampton's side yet again as Brighton came from two goals down to earn a 2-2 draw away to their south coast rivals in the Premier League on Monday.
Virat Kohli received the customised jersey from Southampton striker Danny Ings.
West Brom were relegated from the Premier League on Tuesday after Southampton took a huge step towards keeping their place in the English top flight by beating fellow strugglers Swansea
The draw left Liverpool needing a maximum of four points from their remaining two matches to guarantee qualification for the Champions League via a top-four finish.