Rising star Berahino took his tally to 13 goals for the season by scoring four times in a ruthless display of finishing as Albion coasted into the fourth round of the FA Cup in Pulis' first game in charge at The Hawthorns on Saturday.
The 21-year-old striker put Premier League strugglers Albion in front three minutes before half-time and Victor Anichebe added a second just before the break after Gateshead had been on the front foot.
Berahino netted a further three goals in the second half in a clinical display, with Chris Brunt and James Morrison also on target as Albion ran riot against the Conference National side.
Albion technical director Terry Burton refused to rule out selling Berahino if a big-money bid is received this month amid reported interest from the likes of Tottenham and Liverpool, but Pulis is eager to keep hold of the club's leading scorer.
He told the BBC: "The club has had no offers for Saido Berahino and I've had one day's training with him, and I'm looking forward to working with him. He can score a goal and he's got the balance of a very good player.
"He looks as though he's got the potential to be a good player."
Pulis, who was given a rapturous reception from Albion fans before the game, had planned to spend the entire match in the stands, but was soon barking out the orders on the touchline after his side started poorly against the non-League team.
He added: "There were one or two things I wanted to say and I was too far away from the pitch. It was very unusual for me not to have my cap and tracksuit on the touchline.
"But Gateshead put in such effort and commitment and their supporters should be very proud of their players, coaches and the manager. The longer the game went on the better we got, but it took 15-20 minutes for us to settle down.
"It was good - I got that adrenaline rush before the game and on the touchline I can't stand still. I'm impressed with the club, impressed by the people, and I hope we do well this season and then we can look to put things in place for the longer term."
