Such was England's comfort in the closing stages, Henderson - at risk of missing the semi-final through suspension - could be replaced by Eric Dier as Southgate's side moved on to Moscow.
Kane now heads the table on six goals, two clear of second place Romelu Lukaku, who has been eased out to 10/1 from 8/1.
The win for Belgium saw their odds slashed from 5/1 to 5/2.
The win sends England through to the semi-finals for the first time since 1990.
Gerrard, who was part of the 2006 squad that went out to Portugal on penalties in the quarter-finals, thinks England have a good chance of making history.
Widely written off, including by their own supporters, before the tournament began, hosts Russian Federation know they now stand just two matches from the July 15 final at the Luzhniki Stadium in Moscow.
Meanwhile right back Kieran Trippier, who came to Russian Federation with barely a handful of caps, has established himself as a key player in the team: not just for his pace in defense, but for his countless forward runs and his crossing.
"To come on the back of a game where we had to go to the very depths emotionally and physically in midweek and be able to give what we gave today - control the game, but also withstand the physical test - was a real sign of resilience for a young team, who I think are really maturing in front of our eyes".
"Football can bring connection through a country and I'm delighted we are exciting people, bringing enjoyment and we want to keep it going", added Southgate, who himself has risen to prominence during the tournament with his calm style earning plaudits.
EU package does not cover all demands of Iran
Iran's remaining partners in the JCPOA vowed on Friday to keep the energy exporter plugged into the global economy. Lavrov urged new ways of trading with Iran if Washington restores sanctions against Tehran.
"To be honest for a quarter-final we were quite comfortable and I wasn't expecting that because Sweden area always tough".
The Selecao piled the pressure on their opponents in a gripping second half, but could only drag one goal back through substitute Renato Augusto's header after Fernandinho's early own goal and a sweet strike by Kevin De Bruyne.
Everton's Pickford made his England debut seven months ago and had only three caps before the World Cup but Southgate let him know he would be his first-choice.
"It's a heavy, bitter feeling".
"We are both very risky at set pieces".
Neymar left in tears at the final whistle, refusing to talk to waiting media.
Raphael Varane's towering header gave Deschamps's men the lead before Uruguayan goalkeeper Fernando Muslera's awful error gifted Antoine Griezmann the game-clinching goal.
He said: "I'm hoping, fingers crossed, that we do what we need to do".
"I'm not sure he's always believed that but he's a super talent, a super player and his use of the ball is as good as any defender in the tournament". Sometimes you like to see people be a little bit more positive.