It was clear that Newcastle wanted to make amends for last week's humiliating derby defeat to Sunderland.
But all they had to show for their dominance at the interval was Gouffran's eighth-minute effort, which came after a neat pass from Yohan Cabaye and a good cross from Papiss Cisse.
The opening exchanges had been scrappy but Newcastle had already threatened when Mathieu Debuchy's cross was cleared by Jones.
Cabaye also tested Foster moments later with a well-struck low volley from outside the box, while Mouusa Sissoko broke clear down the left but his pass back for Cisse was scrambled clear by Jonas Olsson.
Ben Foster then dealt comfortably with a spectacular long-range effort from Cisse but he might have feared the worst as the striker seized on a poor pass by Olsson after 35 minutes. Cisse instinctively clipped the ball over the onrushing keeper but his contact was too strong and a fine chance dipped just over the bar.
West Brom did manage to piece a good move together in the closing stages of the half but controversy followed as they were awarded a free-kick on right edge of the box.
James Perch was ruled to have felled Jones after the referee had consulted his assistant but replays suggested the tackle had been clean - and inside the box.
Within moments of the restart Marc-Antoine Fortune tested Rob Elliot for the first time with a long-range effort.
Elliot failed to grab the shot and was lucky to escape as substitute Shane Long seized possession but shot wide, while Romelu Lukaku gave the visitors another warning as he rose highest to head a Chris Brunt corner against the bar.
James Morrison shot wide as pressure continued to build and the hosts' good work eventually paid off just after the hour. James Morrison and Lukaku combined well before the striker turned in the area and picked out Jones, who coolly slid the ball beyond Elliot with a first-time shot.
The game almost turned on its head completely just two minutes later as an unmarked Morrison headed powerfully at goal from a Goran Popov cross but Elliot saved well to his left, while Gareth McAuley won a header from another Brunt corner but failed to hit the target.
Newcastle looked a jaded team and could only break the momentum with a speculative attempt from the halfway line by Gouffran which never concerned Foster.