By the hour mark Gary Bowyer's side found themselves two goals down and a man down, but although they rallied well after Micah Evans pulled a goal back, Fulham ended as strongly as they started to clinch all three points.
On a chilly Lancashire evening Rovers started sloppily with the visitors taking full advantage.
Just five minutes were on the clock when a slide-rule pass from Ronny Minkwitz found Chris David in the area, the midfielder taking a touch before curling the ball beyond Christopher Dilo in the Rovers goal.
Fulham then saw a goal chalked off for offside before the woodwork was rattled just past the half hour, David finding Cauley Woodraw with a cross but the striker's header was tipped onto the crossbar by Dilo.
A deserved second goal for Fulham duly arrived on 38 minutes. Full back Sean Kavanagh taking advantage of space afforded to him to deliver a pinpoint cross to where Omri Altman headed home.
Any chance of a second half comeback looked doubtful when Rovers were reduced to ten men on the hour mark. Winger Fabio Nunes had been yellow carded in the first half for failing to retreat from a free-kick, a second booking followed after a needless challenge leaving Rovers with plenty of work to do.
To their credit however Rovers showed spirit and determination that had been sadly lacking in the first half, pulling a goal back on 63 minutes when from a Robbie Cotton free-kick Micah Evans headed home at the back post to give the home side hope.
Fulham remained a threat going forward with Rovers fortunate once more when Fulham subs Ashley Williams and Charles Banya combined for Banya to slam an effort against the foot of the post, the ball rolling across the six-yard box and away to safety.
Then a neat one-two between Alex Smith and Cauley Woodrow saw Smith lash high over the bar with just Dilo to beat.
With five minutes to go Rovers were within a whisker from grabbing an equalising goal. As Robbie Cotton's corner kick found Jordan Slew at the back post the striker fired goalwards only to see his shot blocked on the line, the woodwork was struck in the ensuing scramble, Kellen Daly unfortunate to see his shot kept out.
The match was eventually put beyond Rovers in added time when a slick passing move saw the visitors add a third, Smith finishing emphatically to give Kit Symons' side all three points.
It had been a better display from Rovers in the last half ahour but their first half display ultimately proved costly.
Speaking to Rovers Player
after the game, coach Colin Hendry said: "We started very poorly, to give away the goal in the manner that we did was not good and after that we were second best.
"We huffed and puffed but we couldn't get a clear chance and they go and get a second goal.
"In the second half we go down to ten men and ironically it gives us a bit of a spur. We get a goal back and we might have got an equaliser. But then they hit the post and score a third and kill the game off.
"For 45 minutes we weren't good enough, in the second 45 it was a better reaction from the team. All in all it hasn't been a good night."
ROVERS: Dilo, Wylie, Beesley, O'Sullivan (sub Slew 61 mins), Paul, Daly (c), Evans, Cotton, Forrester (sub Payne 61 mins), R. Hanley, Fabio Nunes.
Subs not used: Urwin, Maclaren, Osawe.
Goals: Evans 63 mins.
Booked: Fabio Nunes.
Off: Fabio Nunes.
FULHAM: Joronen, Christensen, Kavanagh, Minkwitz, Athurworrey, Grimmer (c), David (sub Williams 63 mins), Altman (sub Banya 67 mins), Woodrow, Islamovic (sub Na Bangna 46 mins), Smith.
Subs not used: Norman, O'Reilly.
Goals: David 5 mins, Altman 38 mins, Smith 90+1 mins