It was Rashford's fifth goal for England in 27 matches.
England fans have had to sit through some turgid friendlies during the past couple of decades, ruined largely by what felt like endless substitutions.
The latter kept a clean sheet against the Swiss but he lacked the assurance of Pickford and the Everton keeper is certain to come back for England's next game, against Croatia on 12 October in the Uefa Nations League.
He almost scuffed a pass into his own goal when trying to find Harry Maguire, and then played James Tarkowski into trouble, presenting Xherdan Shaqiri with an opportunity that he spurned.
The goalkeeper remained with the Potters following their relegation from the Premier League last season, and was the only member of the most recent England squad not to play for a top-flight club.
"I was at Manchester City against Huddersfield and you think: 'Well, OK, it is a game where Foden will play.' But actually, if I am Pep, I have other players, senior players, who I have to get up to speed because they haven't had many match minutes". The 22-year-old produced one moment of quality, threading a pass down the right to Rashford, whose cross teed up Rose for an effort that forced Yann Sommer into his first meaningful save.
After the latest in a long succession of major tournament disappointments, Switzerland coach Vladimir Petkovic is attempting to give his side a makeover.
He was England's focal point in attack during a cagey game against Switzerland, driving his team forward with some driving runs through the middle.
Seeking to match England's shape, Petkovic set Switzerland out in a new 3-5-1-1 system, with Shaqiri playing as a central playmaker - as he had against Iceland, although in a 4-2-3-1 formation - behind lone striker Mario Gavranovic.