England's sub-standard display at Stade Felix-Bollaert suggests they could even struggle to beat Samoa later this month, let alone the Springboks.
Defeats in both games would mean humiliation and an early World Cup exit.
Given that England had their work cut out subduing an American side assembled from clubs such as Park City Haggis, Gentlemen of Aspen and Dallas Athletic, South Africa need not bat an eyelid.
It was the same old depressing story as England somehow managed to 'draw' the second-half 7-7 after leading 21-3, and also inexplicably missed out on securing a winning bonus point.
Poor execution, a lack of composure, bad judgment - all factors behind England's tournament warm-up losses to France last month - let them down once more.
Dallaglio said: "We are going to have to play a lot better against South Africa.
"We have to have a monumental performance next Friday. Physically and mentally, we need to climb several levels.
"We understand what we are trying to do in a game, but if we don't get our tactics right and we play like we did again by shovelling out bad passes and dropping the ball, then against a blitz defence like South Africa's we will be in a lot of trouble.
"It means cutting out the mistakes and playing with fire and passion in the belly you would expect from an England team against South Africa."
Olly Barkley, a convincing fly-half alternative to the injured Jonny Wilkinson, amassed 18 points through a try, three penalties and two conversions, while wing Jason Robinson and flanker Tom Rees also touched down.
Barkley, Rees and Robinson's fellow wing Josh Lewsey all delivered eye-catching displays, yet everything had to be gauged against the quality of opposition, and England were found wanting.
Barkley said: "There were a variety of reasons - poor execution, poor communication and poor strategy.
"And we are aware of how we played. There is no point trying to put any sugar on top of the performance.
"As the tournament goes on, we are going to have to be a lot better than that. Our execution areas were unacceptable."