Southgate, however, at least avoided an unwanted record of becoming England's first boss to oversee four straight losses and also punch the feel-good factor of a race to the semifinals of the World Cup.
"It has not been a perfect week in any way but we ended up with a victory, which is important to move forward," Southgate said.
"We had a tactical problem to solve and a lot of guys played their first 45 minutes of the season," he added of the first half's exhibition.
"It was a difficult task to ask them to play against an excellent team from Switzerland, they kept the ball well and we had to chase too much, there was a lot of room for improvement."
The two teams could have met in the quarterfinals in Russia if the Swiss had not left the competition with a groan to Sweden in the last 16.
England proved too strong for the Swedes before falling short of Croatia in the last four, but it is Switzerland who has so far recovered from the disappointment of the World Cup, the strongest.
Vladimir Petkovic's team defeated Iceland 6-0 to start their campaign in the League of Nations and was the best team during the first 45 minutes at King Power.
"We could have been two or three down at the break," said England defender Danny Rose. "It's great that we can all shout at each other and tell each other to improve as we did."
Southgate insisted after being outscored by most of the 2-1 loss to Spain in their team's inaugural League on Saturday, that England must persist with their intention to improve playing from behind under pressure.
However, it almost cost the hosts the first goal twice in a matter of seconds within the first 10 minutes.
Jack Butland missed a pass that rolled dangerously close to his own goal, and when the elimination of James Tarkowski's goalkeeper was cut off by the Swiss press, Xherdan Shaqiri should have scored instead of hitting the post with his weaker right foot.
Southgate has lamented the failure of England to produce midfielders of the caliber of Croatia and Spain.
But his team, which contained nine changes since the weekend, was struggling to even cope with Swiss passes and moves, as visitors enjoyed almost 65 percent possession before halftime.
A more positive start to the second period for the hosts was rewarded with the only goal in the 54th minute, when Kyle Walker's shot at the far post was swept by Rashford for his fifth international goal.
The importance of enduring Southgate's morale was demonstrated by presenting a series of regular starters, including captain Harry Kane from the bench for the fourth quarter.
Shaqiri was close to a draw, but saw his fierce attack bravely blocked by John Stones as England waited for the victory.