The 24-year-old was at his unplayable best in the 5-3 win over Kosovo at St Mary’s Stadium, scoring once and setting up three of the other England goals.
That came after he scored and set another one up in the 4-0 win over Bulgaria at Wembley on Saturday.
Southgate knows, though, that when words such as “greatest player” are being bandied about, you have to look to those who make a real impact in major competitions.
“How do we benchmark players and where they’re at?” he said. “Definitely winning trophies is important.
“Ultimately, it will come down to performances in the biggest matches.
“No matter how much we think of some outstanding players there have been, in the end they have to have made a difference in the biggest of games for the recognition and the longevity of that.
“He’s without a doubt got the ability to influence those games. We just have to get him into those games for him to be able to do that. You are reliant in football on the team to a degree, and he’s probably going to have those opportunities with his club.
“What we’ve got to make sure is that we help him get those opportunities with England.”
By reaching the semi-finals of the World Cup in 2018, England signalled their intent in that regard. But the whole experience left Sterling at a massive low.
“He did come home a bit sad after the World Cup,” Southgate said.
“There’s no hiding from that. He didn’t deserve to feel that way, but that’s as he was, so it’s lovely to see the recognition he’s getting and how much extra I think that’s bringing out of him.
“Everyone wants to be liked, don’t they? You know how much it means to him, how much he wants to do well. It’s just lovely to see a young man shifting public opinion and establishing himself, at the real top table in the game. We feel very lucky as a group of staff to have him as one of ours.
“We have always picked him. There have been times where he has not played well and we have taken him off and that has been the right thing, but he has always been integral to the way we want to play.
“There are moments where the team were ahead by three or four and it’s possible to look outstanding.
“But in both games this week, at 0-0, a 1-0 down – key moments of the game – he’s just grabbed the game by the scruff of the neck.”
Moreover, when he knocked Joe Gomez off the ball in training this week – an incident which the Liverpool player never heard the end of, it seems – Sterling showed he has the strength to muscle in on the very biggest names in the sport.
“There are obviously some outstanding players around, the Eden Hazards, the Kevin De Bruynes, Cristiano Ronaldo and Lionel Messi,” Southgate said.
“But, he will give himself every chance, because he’s got the drive, he’s got the professionalism, he’s got the ability, he’s strong, he’s physically and mentally strong for such a small-statured lad.”
On the flip side, he also has a caring side and in one final anecdote from the week, Southgate showed why Sterling has become such a valuable asset to England.
“I noticed, over the last 24 hours especially, he spent a lot of time with Jadon Sancho,” Southgate said. “That was probably quite a comfort.
“He can pass things on that I can speak about as the coach, but there’ll be more depth to that if your fellow winger, who has been at the same club as you, and has been through what you’re going to go through, can talk to you.
“I think that’s far more powerful. You can’t quantify what that will have meant for Jadon and it will have helped him settle into the game, I’m sure.”