City took top spot with a 3-1 win over Manchester United on Saturday to reach 43 points but Beever-Jones struck in the second minute to boost the champions in their title defence.
Chelsea did not have it all their own way as West Ham's Honoka Hayashi had a goal wrongly ruled out for offside but an 88th-minute strike from substitute Cuthbert wrapped up the win.
Leicester City's game against Brighton & Hove Albion burst into life in the second half, with the home side taking the lead before the Seagulls eventually snatched a 3-2 win thanks to an 85th-minute goal from Elisabeth Terland.
Tottenham Hotspur striker Bethany England netted for a 1-0 win at bottom side Bristol City to become only the second woman after Arsenal's Vivianne Miedema to score 70 in the WSL. Sixth-placed Spurs have 25 points while City are languishing on six.
The Merseyside derby between Everton and Liverpool ended 0-0 to leave the visiting Reds in fourth spot on 29 points, one ahead of Manchester United, while third-placed Arsenal, who have 34 points, travel to Aston Villa in Sunday's late game.
