Knowing a victory would secure their place in the knockout stages of this summer’s European Championship, England could only labour to a 0-0 draw.
Barnsley-born John Stones hit the post with a towering header from a first-half corner but 60 per cent of possession yielded just one save for former Hull City goalkeeper David Marshall.
“I think it was a frustrating night, we know we can play better,” reflected Southgate. “(We’ve) got to give Scotland credit they defended valiantly and played well.
“We didn’t do enough to win the game but the thing we have to worry about in tournament football after that is we had to make sure we didn’t lose it.
“We know it’s a disappointment for our supporters, but we have to dust ourselves down and move on.”
Captain Harry Kane was again substituted looking well below his best and the introduction of substitute Jack Grealish for his first football of the tournament failed to have the desired effect.
“We needed more runs in behind,” said Southgate, when asked about the form of the man who won the golden boot at the last World Cup.
“I felt (substitute) Marcus Rashford would give us that energy, at that point we had got to try and get the. We have to make these decisions based on what we see.
“The whole team can look at ourselves, that starts with me, we’ve got to be better. We didn’t do enough to win the game, not enough attempts on goal, we have to go away and look at that.
“We’re playing a high level of games under the intensity and to find space was difficult. Scotland played well and we weren’t able to open them up.
“It’s a young team and some of them won’t have experienced a night under this intensity and pressure.”
England’s struggles, though, should not detract from a spirited performance by a Scotland side who looked much better for the return of Kieran Tierney – Leeds United captain Liam Cooper missed out – and an assured full debut for Billy Gilmour.
Former Sheffield United striker Che Adams, having been on the bench against the Czech Republic, missed a good late chance and Reece James headed a Lyndon Dykes volley off the line.
England midfielder Declan Rice warned against an over-reaction to the draw.
“There are games at home we should be winning but credit to Scotland, they ran and defended for their lives,” he said.
“We had our chances to put it away, when John Stones’s header hit the post it felt it might have been one of them nights.
“Scotland nailed their gameplan and nullified Harry Kane, he is a world-class player and we need to get the ball to him to score.
“We need to keep working, (we had) chances in the second half but we don’t need to be too down and negative. With the press and social media there might be an overreaction now but we have to stay positive.”
Scotland manager Steve Clarke was pleased with the way his side responded to Monday’s opening 2-0 defeat at Hampden Park.
“We were unfairly criticised after the game on Monday,” he said. “I know I have a good group of players and they showed that.
“We had chances to win, England had moments as well, you wouldn’t have known which is the favourite. The most pleasing thing was we played when we had the ball and created a number of chances.”
The Three Lions play the Czech Republic, ahead of them on goal difference, in their final group match at Wembley on Tuesday.
The top two teams in the group go through automatically, along with the four best third-placed teams.
Analysis and the Scottish view: Page 2