The former Hearts player, who had previously netted two penalties, broke the deadlock after 61 minutes to cap a second half dominated by the Baggies.
Albion were well worth their victory and can head into the international break full of confidence that they have the ability to finally establish themselves as a top-flight team.
Mowbray's faith in his strikers and midfielders to deliver sufficient goals after a stuttering start to the campaign is starting to be justified.
Bednar is looking more at home in the Premier League with every game, James Morrison is thriving on the flanks and in the middle of the pitch, Albion have the experience and know-how of skipper Jonathan Greening to fall back on.
And Mowbray looks to have picked up a real find in Dutch Under-21 international right-back Gianni Zuiverloon.
Fulham clearly missed the presence of the suspended Andy Johnson in attack and seldom posed a threat as the hosts kept a second successive clean sheet.
They have now lost all three league games away from Craven Cottage and could have few complaints about the result.
The first clear-cut chance fell to Bednar after seven minutes - but he failed to make the most of the opportunity.
Greening's reverse pass picked out the run of Bednar who got goal-side of Aaron Hughes but, after cutting back inside, he clipped his shot straight into the hands of Schwarzer.
Carson, in fine form in recent games, was called into meaningful action for the first time when he held onto a low curling drive from Clint Dempsey.
West Brom defender Jonas Olsson became the first player to be yellow-carded after 14 minutes for bringing down Dempsey just outside his own penalty area.
And Carson was relieved to see Jimmy Bullard's well-struck free-kick fly inches over the crossbar.
Former West Brom star Gera was receiving plenty of vocal support from the Baggies fans but he nearly silenced them on the half-hour mark when he came close to breaking the deadlock.
Dempsey set up the opportunity when he squared the ball back from the by-line into the path of the Hungarian and his precise side-footed effort flew only inches wide.
Borja Valero was booked by referee Phil Dowd for bringing down Gera with a late challenge.
Albion made a positive start to the second half and came close to going ahead within two minutes of the restart.
A deep cross from the left flank by Greening picked out Ryan Donk and the on-loan defender crashed a fierce header against the outside of a post.
Valero had a fierce drive blocked and then Bednar took a chip into the box from Robinson into his stride but volleyed wide.
The pressure was now relentless on the Fulham goal and a close-range header from Morrison was somehow cleared off the line by Danny Murphy.
But Fulham finally cracked after 61 minutes through Bednar's third goal of the campaign.
Greening's corner was turned back across goal by Donk and the former Hearts player was in the right place to stab the ball past Schwarzer.
It was no more than Albion deserved but they were nearly caught napping almost immediately by the Cottagers.
Bullard played a short free-kick to Murphy and his low 20-yard shot was deflected only inches wide,
Bednar had the chance to wrap up the points but Schwarzer did well to block his close-range effort from a Robert Koren centre.
Schwarzer made a fine point-blank save to deny Bednar a second goal from a Koren cross.
Bednar also rattled the woodwork with a rasping drive from fully 40 yards which beat the dive of Schwarzer.
Mowbray: We believe we can stay up
West Brom manager Tony Mowbray insists his side have the self belief they need to compete in the Barclays Premier League after a 1-0 victory over Fulham lifted them to eighth place in the table.
Striker Roman Bednar's second-half effort - his third goal of the campaign - ensured back-to-back wins for the Baggies.
It was no more than Albion deserved after dominating the second 45 minutes and Bednar could have claimed a hat-trick, hitting the bar from 35 yards and being denied at point blank range by Fulham goalkeeper Mark Schwarzer.
Mowbray said: ''We know we played against a decent team today, particularly in the first half and, with Clint Dempsey dropping off the front, they gave us plenty of problems.
''They reminded us of our team at our very best in the first half. They moved the ball around well. Jimmy Bullard and Danny Murphy are top players.
''On the second half performance I think we warranted the victory. We had the better chances, created more chances second half and I'm delighted with the effort we put in to get the points.
''Today's result was important. It keeps the momentum going and gives the players the confidence and belief they can compete in the league and we shouldn't fear whoever we play.''
Victory ensured Albion have enjoyed their best start to a Premier League campaign - 10 points from seven games - but Mowbray is not getting carried away, particularly with a trip to Manchester United next on the horizon.
He said: ''I don't tend to set targets and goals for the players. The next game is the important one and we've got to be ready to go to Old Trafford, make life difficult for them, and see if we can pick something up there.
''It is a bit early to talk about where we might finish. We've just got to look at the next game. We are happy to pick up the points and happy to keep going along.
''We can feel the quality out on the pitch of the teams we are playing this season and we are trying to live with it.
''Our aim is to compete in every game and pick up as many points as we can. Let's keep going, let's compete and see how we get on.''
Fulham boss Roy Hodgson felt his side deserved some tangible reward from the game although most of their opportunities came from outside the box.
He said: ''I did think had we taken one of the many chances that came our way, we wouldn't have been trying again to come back from one nil down.
''We tried very hard, had chances again. It wasn't to be and at the moment we are going through one of those spells where I can't really criticise the players for what they are doing on the field - their effort, quality of play even.
''It is not getting the results and that is a very frustrating situation to find yourselves in.
''It sounds strange but I can't be unhappy with the performance. I am only unhappy with the result. In this league every point dropped is a very important point dropped.''