Turnbull, 22, impressed Southgate in the 1-1 draw at Reading last weekend.
"He commanded his penalty area, came for crosses and dealt with punches well, he's a good kicker and his handling was secure," said the manager.
Schwarzer has been fitted with a plastic shield for his broken thumb while Jones has recovered from a torn thigh muscle. "Whether [Schwarzer] will be ready for Arsenal or not remains to be seen," said Southgate. "We've got excellent cover in those positions, but I didn't think Ross did much wrong at the weekend."