While winning the group was in theory the aim for England all along, first place means the Three Lions will now face the runners-up in Group F at Wembley next Tuesday.
Dubbed the ‘group of death’ before the tournament, England could face either world champions France, 2014 World Cup winners Germany or Euro 2016 winners Portugal, while Hungary also have an outside chance of squeezing into second place.
The Three Lions kept clean sheets in all three group games – a 1-0 win over Croatia, a goalless draw against Scotland and the 1-0 victory versus the Czechs – but have lacked flair up front, with Harry Kane failing to score so far.
But Southgate told ITV: “There is more to come from us, definitely. We’re not fluent but we have moments where we are a good side.
“We haven’t scored from a set play yet. I know people laugh about us only scoring from set plays but they are crucial in big matches and we have to get those nailed.
“We wanted to win the group, we wanted to stay at Wembley and we will wait and see who we play tomorrow.
“They are different sorts of tests but there are a lot of good things we are doing.
“We have managed to get most of the squad into games, 90 minutes for Harry Maguire which was important, Hendo (Jordan Henderson) got 45 minutes which was a real step forward. There were a lot of positives for us.”
Sterling hit a post inside two minutes before heading in Jack Grealish’s cross to win it after 12 minutes.
Grealish was one of four changes and Southgate praised his players for adapting quickly after Mason Mount and Ben Chilwell were forced to self isolate on Monday.
They came into close contact with Scotland’s Billy Gilmour following Friday’s 0-0 draw with Scotland. The midfielder had been diagnosed with coronavirus and it forced Southgate into a late reshuffle.
“I thought we started really well, we were stable with the ball, we had worked on some of the stuff in training yesterday,” he said.
“Credit to the players, some of the stuff we had to adjust without the ball as we had to change what we were doing overnight because of the different personnel.
“It was a huge credit to the players they were as stable without the ball as they were.”
Additional reporting by PA.