The opening goal arrived 10 minutes into the second half as Kyle Walker's impressive cross found Rashford, who had the simple task of slotting home from six-yards out.
With Dele Alli absent through injury and Jesse Lingard dropping to the bench, responsibility for giving shape to England's attacking ideas in midfield fell to Chelsea's Ruben Loftus-Cheek and Manchester City's Fabian Delph.
Southgate's keenness for his players to be competing at the top level is clear, with the 25-year-old the only member of the squad now playing outside the Premier League.
Rashford's lack of game time at Old Trafford isn't helped by his three game domestic suspension after being sent off at Burnley for violent conduct.
But if nothing changes he admits he will have to think long and hard about picking those who have been left stuck on the sidelines for next month's Nations League trips to Croatia and Spain. "It was important then that we solved a couple of the problems they were facing and kept things calm", Southgate said.
"So numerous lads were playing their first 45 minutes of the season so it was a tall order to ask them to go in and play against a really good side in Switzerland, who are an excellent team".
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"We gave them a bit too much respect with the ball".
"I think what is happening at junior level is that there is more emphasis on technical ability".
To make a tough job even harder for the 18-year-old, this summer he saw Guardiola spend a club record £60m on Riyad Mahrez.
"I just want to see him progress, continue to improve and become the great player I know he's going to become".
England struggled to get Loftus-Cheek on the ball in unsafe positions in the first half and when they did, he was unable to do much with it, often showing an inability to turn when presented with the ball with his back to goal.