Makeshift left-back Woodgate had already been booked when he brought down Matt Jarvis for a penalty which was converted by Stephen Hunt. But referee Anthony Taylor did not brandish a second yellow card and Pulis was able to substitute Woodgate after just 19 minutes and reorganise to good effect.
Pulis said: "Jonathan played really well against Tottenham. His first challenge today was a poor challenge (for the booking) and that puts you on edge straight away. We had a bit of good fortune there. If he had sent him off, we probably couldn't have argued about it."
He added: "But when I took him off he said 'that's a great decision gaffer' and that's from someone who has played for Real Madrid and is very much part of what we want to do at this club."
Pulis admitted: "For the first 20-25 minutes, Wolves were very good, much better than us, but for the last 20 minutes of the half I thought we came into the game and had more opportunities.
"In the second half we were the better team. We stepped it up and we were the better team and deserved to win it."
Wolves boss Mick Mccarthy believes referee Taylor should have sent off Woodgate and was also unhappy with the free-kick awarded against skipper Roger Johnson which led to Stoke's equaliser.
He said: "I don't think Stoke should have had 11 men on the pitch. I've asked the referee why Woodgate wasn't booked again but he said it is between me and him.
"Tony Pulis was wise to it and took him off.
"I also think Roger Johnson got the ball when a free-kick was awarded against him and then a deflected goal changes the course of the game."
Source: PA
Source: PA