The Baggies moved into ninth with their second straight win, which came courtesy of first-half goals from Matt Phillips, James Morrison and Darren Fletcher as well as Salomon Rondon's second-half strike, following a day of bad weather in the Midlands.
"It was lovely for us to get four goals for the supporters, it was an awful day and for them to come out and support the team like they did was absolutely fantastic," Pulis told wba.co.uk.
"I am really pleased for them, four goals, good performance and they can go home pretty happy."
Burnley, meanwhile, are still searching for their first Premier League away win of the season having slipped to their first defeat in four matches.
"There was not enough energy, enough fight actually. We were outgunned on every level, more or less."