Gallagher started his first game of the season and got the show on the road with an opener after just ten minutes.
And after Zat Knight was sent off for handling in the area Dickov stepped up to ensure Rovers left with three points.
Fulham once again opted to use Andy Cole as a lone striker and brought Sylvain Legwinski back into the side to cover for the injured Luis Boa Morte.
Blackburn meanwhile went for a front two of Gallagher and Dickov - while Nils-Eric Johansson filled in at left back.
From the first minute the home side looked nervous, and within 60 seconds Blackburn could have been in front.
Dickov burst down the left and fired a cross along the six-yard box, but as Brett Emerton slid in at the far post he was unable to hit the target.
Nine minutes later the visitors took the lead through young Gallagher. Carlos Bocanegra was caught out at left-back allowing Steven Reid to whip in a near post cross.
And Gallagher nipped in between an immobile Fulham back line to stick the ball past Mark Crossley.
The goal came as no surprise and Blackburn could have added a second, only for Johansson to head wide.
But it was Fulham who were gifted the most clear cut chance as Tomasz Radzinski latched on to a Papa Bouba Diop throughball, only to miss the target when one-on-one with goalkeeper Brad Friedel.
The game then began to open up and Reid fired a free-kick over for Blackburn, while Legwinski attempted a hopeful overhead kick at the other end.
Mark Pembridge fired a free-kick over the bar before the break - but it was Blackburn who had completely overawed their hosts and would have been disappointed only to be a goal in front.
After the break Fulham tried to step up a gear as Cole fired over, and then Lucas Neill denied Radzinski a chance at the far post with a superb interception.
Cole forced Friedel into the save of the match on 65 minutes after dispossessing Andy Todd at the back. But his run and shot was tipped wide by the strong hand of the American goalkeeper.
Intense pressure from Blackburn forced Fulham back again in the final 20 minutes and Crossley was forced off the pitch with a groin injury.
Eventually the second goal came virtue of a Knight handball.
As Dickov turned him in the penalty area Knight went to ground and handled. Referee Rob Styles pointed straight to the spot and gave Knight his marching orders for a second yellow card.
Substitute Edwin van der Sar's first contribution was to pick the ball out of his net as Dickov stepped up and sent him the wrong way from the spot.
Brian McBride had come on earlier in the game and had a good chance to bring Fulham back into the match, only to head straight at Friedel when unmarked in the box.
Cole then fluffed a late chance to sum up the Cottagers' afternoon.
Man of the Match: Paul Dickov - The fiery striker posed a constant threat to the Fulham defence and his perfectly taken penalty was just reward for his efforts.