The clean sheet was England's seventh from nine games and they will be looking to finish the group with eight wins.
But there was nothing Oblak could do about Kane's late victor, the cross from former Spurs teammate Walker ideal for him to tap in his 14th goal of the season for club and country.
"Some of that can happen before Russian Federation, some can't because in certain areas there are maybe young players in our system who aren't going to be ready for the next 18 months, two years".
"Tonight highlighted where we are", said Southgate, with the England manager conceding it was "blindingly obvious we could have played better". And I don't mean in November when we play Germany and Brazil in friendlies, if we win or lose two games and everyone gets carried away.
It is hard to recall a less dynamic Wembley display than that and the man in charge of improving matters is eager to find the right formula before next summer. "I think a World Cup in Russian Federation, from what I saw in the Confederations Cup, will be a really good tournament".
Gareth Southgate's men needed last gasp winners to defeat Slovakia in September 2016 and against Slovenia on Thursday night, whereas they also needed a last minute goal to draw against Scotland in June.
"In the last 10 minutes, they were getting exhausted and we created a few opportunities - I was happy to finally put one in the back of the net at the end".
If that doesn't occur then the trip to Lithuania will certainly be a lot more stressful than necessary.
"It is an unbelievable feeling to score, no matter what the time, and it's great to go through to the World Cup".
Hart is set to earn his 74th cap in England's penultimate World Cup qualifier, which would move him clear of Gordon Banks and one short of David Seaman's 75.
Southgate said: "Slovenia defended resolutely and we weren't as sharp as we'd like to be". He's been fantastic for us but you don't want to go into a tournament with everything on Harry Kane. Kane poked home Kyle Walker's cross in the fourth minute of stoppage-time. We want to stay unbeaten, we want to go and win the game. You can't boil Spain's 2010 World Cup victory and Germany's 2014 triumph down to a single factor, but it's impossible to ignore the fact that both depended upon the core of a cohesive, intelligent club side (Barcelona and Bayern Munich, respectively).
"We should enjoy this. We should be proud as a team and as a nation", the Spurs striker said. There are some big countries around the world that are struggling.