Pulis introduced Michael Kightly and Dean Whitehead in the second half at The Hawthorns and the pair combined for the latter to score the only goal of the game in the 75th minute.
It sealed Stoke's first away victory in 17 Premier League games and they have now taken 13 points from their last five matches following the 2-1 win at Newcastle in midweek when sub Cameron Jerome set up the winner.
Pulis said: "We've been fortunate with the substitutes during the last two games.
"We changed it on Wednesday and Cameron came on and made one and scored a great goal for us.
"Then that has happened today with the guys combining for the goal. Sometimes that happens for you.
"I've not turned into a genius. It's just the little bit of luck you need."
Pulis was delighted to see his side finally pick up three points on their travels.
He said: "Obviously the monkey is off our back. We've had four draws away from home this season and we could easily have won a couple of those games and it hasn't just fallen for us.
"It is always nice to get the first win and I'm really pleased."