"There's a lot of work to do still", said Southgate.
"We created so many clear-cut chances, especially in the first half, and were in total control in the second half", Southgate told BBC One.
England had a legitimate case for a spot-kick at the other end when Kane was bundled over in the box as Alli headed against the crossbar.
"It's massive. I'm so proud of the lads", he said.
There were bonuses elsewhere for Southgate, especially in the performance of Tottenham right-back Kieran Trippier, who gave a consummate performance as a defender but also going forward, where he was a constant threat in open play and set-pieces.
South Africa 2010 started badly and never really got going for Fabio Capello's team before they were blown away by Germany in the round of 16, while Brazil 2014 was a disaster of two defeats and a draw against Costa Rica before finishing rock bottom of the group. Ashley Young delivered instead - you know, one of those players out on the wing who can cross well - and John Stones met it with a ferocious header.
Southgate's men will have been frustrated by their profligacy, Lingard latching on to a ball over the top and knocking it past Ben Mustapha, only to see it trickle onto the post and wide. "I'm delighted for him as we'd be asking questions about him not scoring in tournament football if he hadn't scored".
Kansas Sheriff’s Deputies Killed In Confrontation With Inmate
Gov Jeff Colyer was briefed on the incident as he arrived to a congressional forum hosted by the Kansas City Chamber of Commerce. Looking out a window, he saw three people on the ground in a parking lot behind the building, Garcia told the Kansas City Star .
Kane - who has now scored 15 goals in 25 caps - was twice wrestled to the ground in the box without the video assistant referee awarding a penalty. On occasions Trippier found himself nearly in line with England's attacking players while trying to shut down Tunisia and it did find success in causing mistakes. I think England rolls to an easy tournament opening win on Monday, so I'll back the Three Lions with a -1.5 handicap.
"A few corners, they were trying to grab, hold and stop us running".
'We've done for a while since gaffer has been here, good togetherness, ' he said.
"Whilst you would have heard all the interviews of having "played well, being over it and building to Panama", it would have been hard over the next few days".
England enters the 2018 World Cup saddled with baggage and no expectations to fill those bags with.
They found one in the form of a header from Harry Kane, whose uncanny ability to find himself in the right place at the right time once again pushed his team beyond the brink.
The second half started much the same as the first, with the English pinning the Tunisians back inside their own half.