England shipped two late Leigh Griffiths free-kicks in Saturday's 2018 World Cup qualifier at Hampden Park, only for new captain Harry Kane to rescue a point by volleying in a 93rd-minute equaliser.
Southgate is trying to rebuild England's confidence after the shock of last year's Euro 2016 humiliation by Iceland and said his players' perseverance in the face of improbable odds was a sign that they are on the right path.
"I think it's a huge moment for the team," said Southgate, who took his squad to a military training camp on a team-building exercise last week.
"The questions around us centre on character and the ability to withstand events that go against you. That's what we've got to show.
"We have to be a team that are never beaten. We have to be a team that, maybe the clock runs out, but you never, ever stop.
"Today we've done that. Is it what we wanted when we turned up here? No, it isn't. But the chain of events is remarkable really."
Kane's goal was the Tottenham Hotspur striker's sixth in 18 international appearances and came at the end of his first game under Southgate, after injuries had kept him out of England's six previous matches.
"It was an important moment for the team and an important moment individually for Harry," Southgate told his post-match media conference.
"I've a feeling we might have been sitting here talking about how long it was since he last scored for England if he hadn't scored.
"It was brilliant for him that in a pressure moment, he executed a skill that he made look a lot easier than it actually was. I think it's a significant goal for many different reasons."
'Scotland's best free-kick'
On a sun-kissed Glasgow afternoon, England took the lead in the 70th minute when substitute Alex Oxlade-Chamberlain's left-foot strike beat home goalkeeper Craig Gordon's flimsy attempt at a save.
It was the cue for Griffiths to take centre-stage, the Celtic striker arcing a 25-yard free-kick inside the right-hand post in the 87th minute and then curling a shot into the other side of the goal three minutes later.
England goalkeeper Joe Hart, whose Manchester City future remains in limbo following a loan spell at Torino, got nowhere near either shot, by Southgate absolved him of blame.
"I thought he was extremely composed with the ball at his feet and his focus through the week, he's been one of the outstanding leaders in the group," said Southgate, whose side remain top of UEFA qualifying Group F.
"I'm sure he'll be disappointed to concede any goal. We'll have a look in more detail, but you've got to credit Leigh Griffiths with two outstanding strikes."
Griffiths, 26, had never previously scored for Scotland and his goals drew rich praise from manager Gordon Strachan after a result that kept Scotland in fourth place in the pool.
"I've seen Scotland's best ever free-kick and then I've also seen Scotland's second best ever free-kick," Strachan said.
"And then a noise after that, that I have never heard anything like before. So these memories will be with me. I will look at the players and tell them you have to keep coming back and taking knocks like that.
"It is annoying for them. I can't be annoyed. It is annoying for them to do so much and have a result that could have been one of the best results ever."
Kane's goal denied Scotland a first win over their old enemies since 1999, but Strachan could not fault his players.
"You don't often see people push themselves to another level and that is where they pushed themselves today to get a result for this country."