Sweden beat Italy 1-0 to gain the upperhand after their World Cup playoff first leg on a chilly night, while a young England side held Germany goalless and Belgium drew 3-3 with Mexico despite Romelu Lukaku's double but France eased to a 2-0 victory against Wales.
Stones headed England in front in the eighth minute, losing his marker to angle home a Kieran Trippier corner before captain Kane made it 2-0 in the 22nd minute with his third goal of the tournament, an emphatically struck penalty after Jesse Lingard was pushed over in the box by Fidel Escobar.
On a seismic day in the history of Chelsea, Romelu Lukaku scored the winning goal in a 3-2 FA Cup fifth-round victory
Chelsea kept up their pursuit of the Premier League's top three clubs with a 1-0 win away to London rivals Crystal Palace on Sunday, thanks to a strike by N'Golo Kante early in the second half.
After more goals came from set-pieces in this World Cup's group stage than any other, dead-ball strategy has continued to have a huge impact in the knockout phase, with five of the 11 goals scored in the quarter-finals coming from free kicks or corners.
Chelsea enjoyed a somewhat fortuitous return to winning ways when they beat Aston Villa on Saturday but their fragile self-confidence will face a far more severe test at Dynamo Kiev in the Champions League on Tuesday.
Tottenham Hotspur climbed to third with a 3-1 win over Southampton while Chelsea hung on to fourth spot, above Arsenal on goal difference, despite a 2-1 defeat at Wolverhampton Wanderers.
Southgate's formation and tactics have been hailed as progressive and modern, there is also something traditionally English about the team's strengths.
A round-up of Saturday's action in the Premier League.
Manchester United striker Romelu Lukaku defended Jose Mourinho's confrontational style of management, saying that the Portuguese deserves respect for not hiding his true feelings like several Premier League bosses.
Images from the English Premier League matches played on Saturday.
Images from the UEFA Champions League matches on Tuesday night.
Premier League Swansea City fell victims to the first major giant-killing of the FA Cup third round on Sunday when they lost 3-2 away to League Two (fourth tier) side Oxford United.
Arsenal's Pierre-Emerick Aubameyang struck a vital late goal to seal a 3-1 win over Valencia on Thursday and give his side breathing space for the second leg of their Europa League semi-final in Spain next week.
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.
First-half goals by Danny Welbeck and Alex Iwobi earned Arsenal a first win in four league games to revive their Premier League title hopes in the early kickoff at Goodison Park on Saturday.
Images from Tuesday's action in the UEFA Champions League.
IMAGES from soccer matches played in the EPL, La Liga and Serie A on Tuesday.
Images from the English Premier League matches played on Saturday
News of all that's transpired on and off the football field
IMAGES from the English Premier League matches played on Wednesday
Images from the English Premier League matches played on Saturday
Swansea City's hopes of salvaging their place in the Premier League suffered a potentially grievous blow as Ryan Fraser's deflected strike earned Bournemouth a 1-0 win on Saturday to secure another season of top-flight football for the Cherries.
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.
News of all that's transpired on and off the football field
City's Aguero doubtful starter for Watford trip
Chelsea found no chink in Everton's defensive armour as the Premier League champions were held to a frustrating 0-0 draw at Goodison Park on Saturday.
News of all that's transpired on and off the football field.
News of all that's transpired on and off the football field