The Swiss took a first-half lead through Breel Embolo but Luke Shaw levelled before the break and Kane grabbed the winner from the spot late on. Kane’s strike made it 10 wins and three draws against the Swiss since a World Cup qualifying defeat in 1981.
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The hosts played the second half without names on their shirts to highlight how people living with dementia lose precious memories.
The visitors looked more organised and more energetic in the first half and went ahead when England centre back Ben White, a late inclusion in the team after John Stones was injured in the warm-up, misjudged an excellent Xherdan Shaqiri cross and Embolo nodded in.
England keeper Jordan Pickford made two excellent saves to deny Granit Xhaka and Fabian Frei as the hosts created almost nothing until first-half stoppage time, when the impressive Conor Gallagher sent over a low cross and Shaw crashed home a superb shot from the edge of the box.
Second-half substitutions disrupted the flow of the game but England were awarded a penalty after a VAR check as Steven Zuber handled Ghehi’s header inside the box.
England captain Kane stepped up to convert the spot kick and Gareth Southgate’s side held on for the win.
It was the striker’s 49th international goal which means he is level with Bobby Charlton and only four goals behind all-time scorer Wayne Rooney.
“It’s big year ahead, a chance to try some different systems, different formations,” Kane said. “There’s always room for an improvement but it was a good win to start the year.
“We rotated the squad a bit, some new faces made their debuts, which was great, and these are the games where we need to try stuff. We haven’t got a lot of time so all this information is key.”
On his goal and his position in the England scoring charts Kane added: “Penalties are a great way of getting on the scoresheet, I practice and work on them a lot.
“It’s amazing company to be with. I’m super proud to be doing that. It’s a big year ahead to get more caps and more goals.”
Information from Reuters was included in this report.