Fulham's play also carried a threat and former Baggies favourite Zoltan Gera, who received a generous ovation when his name was called out at the start, had a gilt-edge chance to score against his former club.
A determined Dempsey slid in to beat Gianni Zuiverloon to the ball and he pulled a pass back to the unmarked Gera, only for the midfielder to angle a side-footed shot narrowly wide.
West Brom upped the intensity after the break and Donk headed a far-post cross from Greening against the upright from a tight angle.
The home crowd urged their side forward and Bednar controlled a Paul Robinson pass only to shoot wildly over with his second touch.
Hodgson laments Fulham's lack of luck
Fulham were clinging on and Danny Murphy somehow cleared a close range James Morrison header off the line.
But, almost immediately, the visitors buckled as the home side made their pressure count.
Donk had a shot blocked in a crowded goalmouth as Fulham failed to clear a corner and the ball found its way to Bednar, who poked in a shot.
The lead would have been short-lived, but hardworking Baggies midfielder Robert Koren flung himself at Murphy's strike to deflect the effort just wide.
Fulham injected more urgency into their game and Dempsey headed over when unmarked before Seol curled an effort wide.
Koren had an angled shot saved and Bednar struck the bar with a 35-yard effort as West Brom tried to finish Fulham off.
But they did enough for the win and achieved their first back-to-back wins since beating Charlton and Everton on 19 March and 3 April 2005, respectively.
West Brom boss Tony Mowbray: "The players work extremely hard. We've had seven games now and we have been in every game. We've competed and this game was no different.
"The players are a honest and hard-working bunch with quality and we are delighted with the win."
He added: "Roman (Bednar) was a bit disappointed early on in the season to be on the bench but he has done extremely well."
Fulham boss Roy Hodgson: "It was always going to be a tight game and both teams played well.
"The away form is a concern because it is not easy to go to another ground and get a positive result.
"If we continue playing the way that we are then the results will come. We have to hope that the tide will turn."
West Brom: Carson, Zuiverloon, Donk, Olsson, Robinson, Morrison (Cech 79), Koren, Greening, Borja Valero, Miller (Moore 64), Bednar. Subs Not Used: Kiely, Hoefkens, Barnett, MacDonald, Pele.
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