The Three Lions will take on their old rivals at Wembley next summer after the Scots sealed progression to the delayed tournament and the England boss is licking his lips at the prospect of the game.
Southgate told BBC Radio 5 Live: "What a game. That makes our lives more complicated for sure.
"It will hopefully be a fantastic occasion with fans, but that's the one thing none of us can control."
Southgate revealed that he was "delighted" to see Scotland triumph in their play-off against Serbia and congratulated boss Steve Clarke on the achievement.
He said: "I'm delighted for them.
"I'm particularly pleased for Steve Clarke. He's a great guy and has done a great job. That will be a humdinger of a game.
"They've got a team that is improving quickly with a lot of players who have emerged in the last two or three years. The likes of John McGinn, Scott McTominay, Kieran Tierney, Andrew Robertson."
Southgate was speaking after England's 3-0 friendly win over the Republic of Ireland where he was full of praise for midfielder Jack Grealish.
He said: "Jack was very good.
"He's in really good form and he's playing in the areas we want him to play, not coming too deep and affecting the game in the final third. Some of the interplay down the left was really exciting."