So underwhelming was this encounter that fans seemed more excited about paper planes landing on the pitch, with England offering precious little to shout about until captain Kane directed home to secure a stodgywin in stoppage-time.
England is close to securing qualification for the 2018 World Cup, and can lock up its bid to Russian Federation with a win over Slovenia Thursday.
The England manager regards the 21-year-old as a crucial player and an nearly certain starter at next year's World Cup, but will warn him that such acts of indiscipline could cost him his place or damage the team's prospects.
Group winners qualify automatically for the finals in Russian Federation.
With his third hat trick in the campaign, Poland striker Lewandowski became the most prolific scorer in a European qualifying campaign with 15, one ahead of Cristiano Ronaldo.
This was nearly an excellent result for Slovenia, who drew with England in the return fixture in Ljubljana.
"We know we have to get better but that's for tomorrow", Southgate said. "And then it's, 'We come into training, am I a shoo-in to get in the team or do I have to push a bit harder because I have a couple snapping at my heels?"
"Tonight, we go to the World Cup and that's fantastic".
"Joe Hart [made] big saves at important times and that's where his experience has come through".
"Up to this point, they haven't conceded many goals at all so it's a great challenge for us to change that stat".
However, the 24-year-old produced the goods again, scoring his 14th goal in nine games for club and country, to snatch a 1-0 win and spare England's blushes.
In a largely turgid affair, England missed the suspended Dele Alli and the injured Jamie Vardy's ability to move defenders around, with Slovenia happy to sit deep and soak up pressure.
What time is England vs Slovenia kick-off?