Goals from Patrick Kluivert, James Milner and Alan Shearer were enough for United while Geoff Horsfield scored a consolation for West Brom, who had Darren Purse sent off in the second half for a second yellow card.
It was Newcastle's third Premiership win in a row and fourth in all competitions, but the result left embattled Gary Megson still searching for his first win of the season, though he clearly had the backing of the Baggies' travelling fans.
Souness kept the same team that ended United's 32-year wait for a win at Southampton, though he tinkered with the formation and Craig Bellamy played up front with Shearer and Kluivert dropped off behind the front two.
Megson, meanwhile, restored Kanu and Robert Earnshaw to the starting line-up, having left the pair out of Tuesday's Carling Cup defeat at Colchester United.
Bellamy had the best of the early chances, but glanced a header just wide of Russell Hoult's goal from the lively Lee Boywer's cross in the 17th minute.
Within seconds Earnshaw looked like getting the better of Robbie Elliott as the pair raced after a long ball forward. The Newcastle defender was beaten for pace, but used his strength against the Wales striker to good effect and shepherded the ball to safety.
Referee Mike Riley was the centre of attention in the 24th minute after he turned down what looked to be a clear penalty when Purse, already booked for a foul on Bellamy, blocked a Shearer shot with his hand.
Bowyer had another penalty appeal turned down after he tangled with Thomas Gaardsoe in the box. An infuriated Megson, once a United player, signalled to the fourth official that the Londoner had dived.
Bellamy sent two chances wide of goal as the crowd became restless, before Kluivert twice went close in the 40th minute, having two goalbound shots blocked with the second leaving Riccardo Scimeca prostrate on the pitch.
Bowyer almost broke the deadlock on the stroke of half time, but shot into the side netting from 12 yards after Olivier Bernard's ball had found him arriving late into the box.
Megson replaced Earnshaw with Horsfield for the second half, two minutes into which Albion's Neil Clement went close from a 20-yard free-kick.
Bellamy was replaced by Laurent Robert ten minutes into what was a scrappy half. Seconds after the substitution Bowyer almost broke the deadlock but his deft chip rebounded to safety off the frame of the goal.
In the 64the minute the Baggies went down to ten men when Purse received a questionable second yellow card for a foul on Kluivert, and in doing so became the first-ever player to be sent off in a league fixture between the two teams.
In the 70th minute West Brom fell behind when Kluivert notched his fourth goal for his new club. Hoult could only push away a Robert cross as far as the Dutchman, who slammed a right-foot shot into the net.
Robert, making up for lost time, was then unlucky not to score himself, seeing a shot come back out off the underside of the crossbar.
Goalscorer Kluivert then made way for a second winger, James Milner, in the 77th minute and within seconds the youngster was on the scoresheet, literally walking the ball over the line from a Jenas pull back, for his first competitive United goal.
Shearer made it 3-0 in the 86th minute from close range with West Brom's defence in disarray. A minute later Horsfield pulled one back, bravely heading home from a fine Andy Johnson cross.
MAN OF THE MATCH: Lee BowyerLee Bowyer was in the middle of everything for Newcastle United, with defensive-minded central midfield partner Nicky Butt allowing him to play his natural attacking game. Bowyer created plenty of chances, and had a couple himself. He also put hisfoot in when needed and dropped deep when Butt broke forward. Their partnership, while in its infancy, looks very promising for Newcastle.