The Cottagers went in front early on when Scott Carson pushed Zoltan Gera's shot on to the post, only for it to bounce back against the goalkeeper and over the line.
But the Baggies were soon level through Youssouf Mulumbu before Marc-Antoine Fortune's goal turned things around in the 40th minute.
The result means the Baggies have now gone seven games unbeaten in all competitions as their impressive start to the season continues.
Fulham, meanwhile, are still looking for an away league win, having not taken three points from a fixture on the road since the opening day of last season.
West Brom made the more positive start, with Chris Brunt drilling a low free-kick that Mark Schwarzer was equal to.
Moments later Fortune chested down Brunt's cross in the box, turned and hit a shot that deflected behind for a corner. It was met by Jonas Olsson, who was unable to find any accuracy with his header.
Clint Dempsey hit back at the other end with a speculative effort that went wide and in the ninth minute, the visitors were ahead thanks to a slice of good luck.
Gera collected a pass from Moussa Dembele and unleashed a strike which Carson did well to tip against the post.
Unfortunately for Carson, the ball bounced back off the woodwork, against him and in, and although Gera made a point of not celebrating against his old club, it will go down as an own-goal.
Another former Baggies player, Diomansy Kamara, could have made it 2-0 soon after but saw his effort go just the wrong side of the post, and Gera then shot over the bar.
Just after the quarter-hour mark the hosts were level, though, as Brunt's through-ball evaded Fortune but was controlled by Mulumbu, who slotted it past Schwarzer.
Brede Hangeland then gave the ball away in a dangerous area but was swift to reclaim it from Jerome Thomas, and Olsson was on the end of another corner with a looping header that Stephen Kelly cleared off the line.
The home side were back in the ascendancy and with five minutes of the first half remaining, they took the lead.
Thomas picked out Fortune from the by-line and the Frenchman sidefooted the ball beyond Schwarzer and into the net.
It could have got even better for Roberto Di Matteo's men before the interval but James Morrison shot well off-target and Paul Scharner's header went into the side-netting.
West Brom emerged for the second period looking eager to extend their advantage and Morrison tried his luck again, with a similar outcome.
Fortune then attempted to take advantage of another mistake by Hangeland, but over-ran the ball as he advanced towards Schwarzer.
Di Matteo made a double substitution, withdrawing Fortune and Olsson for Roman Bednar and Pablo Ibanez, and Gabriel Tamas connected with a corner but could do little to send it in the right direction.
Andy Johnson came on for Fulham in place of Kamara to make his first appearance since January, but it remained one-way traffic, with Bednar hoisting the ball over Schwarzer and in only to be flagged offside.
Morrison then received the ball by the far post but lost his footing as he tried to poke it past Schwarzer, and Scharner's shot drew a good save out of the Australian goalkeeper as the hosts continued to pile on the pressure.
Baggies fans cried out for a penalty when Thomas appeared to have his shirt pulled by Kelly, but referee Kevin Friend waved play on.
West Brom then sprung an attack direct from Carson's long kick, with Morrison racing on to it and prodding the ball across Schwarzer but wide.
Fulham had a glimpse of a chance when Carson flapped at a cross, but the ball was cleared away.
Moments later Johnson narrowly failed to get a touch on a delivery that flew across the box.
The former Everton frontman was then unable to get a shot in while tracked by Ibanez and the hosts remained solid up to the final whistle to claim the victory.