Harry Kane’s hat-trick was the highlight of a dominant 5-0 win at Wembley and put England a point away from reaching next year’s World Cup in Qatar.
Reports before the game suggested Southgate was expected to sign a new two-year deal to take him through to Euro 2024, but the boss insisted it was not his focus.
He told ITV: “It’s an irrelevance for me. I’m under contract, I’m not out of contract, and I’m very well backed and supported here. My focus is getting this qualification done and everything else can wait.
“We need a point, we are not done, and it would be wrong for my attention to be anywhere else than that. My job was to get this team to a World Cup and then we can look at everything else after that.”
Skipper Kane also told ITV the players want Southgate to extend his stay.
“I personally, and the lads, love Gareth,” added Kane. “We want him to stay but it’s his personal decision and we’re backing whatever he does.
“We’ll enjoy tonight, he’ll enjoy tonight. Let’s get one more win on Monday and then we can enjoy a fantastic year.”
England need a point in San Marino on Monday to book their place at next year’s World Cup.
Southgate added: “The first half was fantastic, the best we have played for a long time, but it’s always difficult to maintain that when you are so far ahead. We were still professional in the second half, but unfortunately we couldn’t give the crowd more goals.
“We are best when we have an edge. We didn’t produce what we should against Hungary and we wanted to put that right and we had a game where we had a chance to put us on the verge of qualification for a World Cup, simple as that.”
Kane passed Wayne Rooney’s record of 37 competitive goals for his country with his hat-trick to earn praise from Southgate.
“I thought he gave a brilliant centre-forward’s performance,” said Southgate. “He competed, he held the ball up, he played other people in and his finishing was outstanding. His all-round game was excellent.”
Tottenham striker Kane moved on to 44 England goals – level with Jimmy Greaves – and believes the players answered their critics after a 1-1 draw with Hungary last month.
He said: “Whenever you put on an England shirt you want to do everyone proud – your friends and family, the fans – and we may have had a little point to prove after the last game. We didn’t perform to the standards that we know we can.
“It was an incredible performance. The pressure was on a little bit and we needed to get the result. I can’t credit the lads enough and it’s a great way to get closer to qualifying.”