Pulis brought on Cameron Jerome against Newcastle in midweek and he set up a goal for Jonathan Walters before scoring the winner in a 2-1 success.
The Potters manager then introduced Michael Kightly and Dean Whitehead against the Baggies and the pair combined for the latter to score a 75th minute goal.
Pulis said: "We've been fortunate with the substitutes during the last two games.
"We changed it on Wednesday and Cameron (Jerome) 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 for Stoke to finally pick up three points on their travels.
He said: "Obviously the the ***** 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."