Pulis revealed he had made the decision to play the former Tottenham, Real Madrid and England defender several days earlier.
"I decided about a week ago," Pulis said. "I thought this would be the right game for him, given his experience."
Woodgate came into the heart of Stoke's defence with Robert Huth switching to full-back in place of Andy Wilkinson, who was not included in the squad.
Rory Delap - replaced by Whitehead in the starting XI - had not travelled with the group and Pulis confirmed both he and Wilkinson were nursing injuries.
"Rory has got a slight hamstring strain, so we decided not to bring him," Pulis said.
"Wilko has actually broken a toe. Although he has trained for the last couple of days, if he had got a knock on it, he could have been out for four or five weeks and we didn't want to take that risk.
"Robert has played at right-back before and although it is not his favoured position, he is not bad at it."