The Three Lions were pushing for a return to winning ways after losing 2-1 to Spain over the weekend, while Switzerland thumped Iceland 6-0 in their UEFA Nations League opener.
England didn't get off the best start in the first half after making nine changes from their game against Spain, but Southgate made a few tweaks for the second half before Rashford's 54th-minute victor.
Rashford's strike put some gloss on what was a less-than-convincing performance by the World Cup semifinalists, which was typified by the jitters of goalkeeper Jack Butland.
"Harry falls in the category in which we have several players where we have to watch how much they play", said the Three Lions boss.
Not surprisingly the upheaval resulted in what Danny Rose described as an "embarrassing" first-half display as the players struggled to find their rhythm.
A lacklustre opening 45 from England saw the first half come to a close goalless, with the visitors probably feeling like they should have been in the lead.
"We didn't get hold of their pivot in the first half, [Granit] Xhaka was always available and their back three were finding him too easily". They recognise when they want to be better.
We know all about the influx of foreign players into the Premier League and how the selection pool is becoming ever smaller to pick from. "So I thought this is a good opportunity to do it".
'It's important they have a view, they are intelligent football players, they should have a view of what's going on.
Southgate, meanwhile, said of the row between the players at half-time: 'We expect that, we encourage it. We also know that, on Saturday against Spain, we could have feasibly ended with a draw which would have looked different as well. "I think for the teams that went towards the end of the World Cup it's a really hard period because so many players have missed pre-season and you're not really sure what physical condition they're in".