Kane became England’s highest ever scorer at major tournaments after Jordan Henderson opened the scoring, with Bukayo Saka superbly wrapping up a comfortable victory in Sunday’s contest at Al Bayt Stadium.
The reward is a crunch quarter-final clash against holders France next Saturday.
Southgate’s side will need to take their performance up a level given some of the problems Africa Cup of Nations winners Senegal were able to cause in a first half that ended with a quickfire England double.
“It was really tough, knockout games are never easy,” he said afterwards. “I think we’ve showed great maturity through the tournament, and 3-0 against a really good side, credit to the boys, the mentality was top from the beginning and we took our chances when they came.
“We feel good, we’ve got good runners, good players getting forward, and it’s solid at the back, three clean sheets in a row is really important. A really good day for us.”
Regarding his goal, which leaves him one behind Wayne Rooney’s all-time England scoring record, Kane said: “I had so much time, the ball just sat up nicely, the connection was perfect.
“I had a chance before that which I should have done better with, so it was nice to see that one go in. I’m a striker, so whenever I score I’m always happy.”
Phil Foden was full of praise for the influential Jude Bellingham, who played key roles in the first two goals for England.
Foden said: “I don’t want to big him up too much because he’s still young, but he’s one of the most gifted players I’ve ever seen.
“I don’t see a weakness in his game, I think he has everything, and I’m sure he is going to be the best midfielder in the world.”
Bellingham himself was keen to speak about his fellow midfielder Henderson, who he said deserves respect.
The 19-year-old said: “It was a brilliant performance (by Henderson) today. I’ve seen some of the rubbish that was said about him playing. It’s ridiculous.
“He’s so underrated technically, and he’s delivered again in a big game with a massive goal. I think it’s time he gets a bit of respect.”
Manager Gareth Southgate emphasised how pleased he was with “the ruthlessness of our execution, the way the team worked off the ball and without the ball”.
And he added: “In the first 25, 30 minutes, Senegal had a lot of energy, they pressed well, their shape was good, and really our best opportunities came from winning the ball from them.
“I thought Jude and Hendo in particular were outstanding in that aspect of the game.”