Raheem Sterling's first tournament goal – and England's first at Wembley in front of fans since November 2019 – was enough to secure a 1-0 win in sweltering conditions.
It was an ideal start to Group D but Southgate knows the challenges continue, with Scotland next up at Wembley on Friday night.
"This is the big challenge for us now because of course we're delighted with the win and the players are happy," the former defender said.
"It's a positive step in terms of qualification, but one of the hardest things in sport is to come back to that point you were at before the kick-off for the next game.
"You can get a bit of a glow of the win and a bit soft, a bit puddingy when you poke your finger in the belly.
"We've got to make sure that we're ready for what is going to be an incredible game against Scotland and a huge challenge.
"We should enjoy tonight because I think you have to enjoy your wins, especially in tournaments, but focus now.
"Our first objective is qualification. We haven't done that yet and we have a very important game for us and our fans on Friday night."
Sterling's goal was enough for England to get their campaign started with a victory and Southgate was pleased to send the fans home happy.
England will return to Wembley to take on Scotland after a few days of excited chatter and growing expectation.
"I think the pressure is what we decide it to be," Southgate said. "We've talked about this as a group.
"We have opportunities to be first to do things. I didn't realise today was our first win in an opening game, so that's something they've managed to achieve that no other team has done."