Berahino, 22, had said he would not play under Albion chairman Jeremy Peace again after the club rejected two deadline-day offers from Tottenham.
But he made his first start since at Villa, poking home a first-half strike.
"It's over now. As a club we move on and he's a very good player," Pulis said after the game.
"This game was made for Saido. He's not played a lot of football and looked tired as the game went on, but he was desperate to play and he has the quality that we want and need.
"Everyone makes mistakes - in a few years' time he will look back on the situation and think he should have done better."
England Under-21s forward Berahino returned to the Baggies squad last week after a three-game absence, coming off the bench in the 0-0 draw with Southampton.
West Brom manager Tony Pulis says they have moved on from Saido Berahino's threat to strike after he scored the only goal in their win at Aston Villa.