The Potters manager introduced Michael Kightly and Dean Whitehead against the Baggies and the pair combined for the latter to score a 75th-minute goal.
It sealed Stoke's first away victory in 17 league games since January when they triumphed at Blackburn and Pulis said: "We've been fortunate with the substitutes during the last two games. I've not turned into a genius. It's just the little bit of luck you need."
He added: "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."
Pulis was delighted for Stoke to finally pick up three points on their travels.
He said: "Obviously the 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."
Albion have now lost their last two games and slipped from third to fifth in the process but Baggies head coach Steve Clarke refused to blame defender Jonas Olsson for the goal even though it was his clearance which fell to Kightly before he crossed for Whitehead to break the deadlock.
Clarke said: "To be honest it was the kind of game we thought it might be. The two teams cancelled each other out. It was a game of few few chances. We spoke before and said it was probably a game of one chance.
"We said the same at half-time and unfortunately for us the chance fell to Stoke. Jonas made the decision to take the ball out to the touchline. For me, it's a defender's decision.
"He is on the pitch, he has to make that call, he made the right call."
Source: PA
Source: PA