Dallas struck from outside the box in each half with further goals from Alan Judge and Ramallo Jota adding to Fulham’s misery.
The three points lifted Mark Warburton’s Brentford to fifth in the table while Fulham remain in 20th – eight points clear of the drop zone with six games remaining.
Fulham have struggled at Craven Cottage recently having lost 5-1 to Bournemouth followed by painful 3-0 home defeat to Leeds.
However, they won their first match in five games in their last outing at Huddersfield thanks to a 2-0 win and they started confidently against Warburton’s play-off chasers.
Encouraged by the vocal home support they powered forward and created the first chance after three minutes.
Danny Guthrie, the midfielder on loan from Reading, picked out Alexander Kacaniklic who tricked his way past Toumani Diagouraga but his powerful, angled shot was beaten away by visiting keeper David Button.
Brentford have lost only one of their last five games and their success this season is largely down to their free-flowing attacking football.
The flip side is they have conceded more goals than any team in the top eight of the Championship.
But they always carry a goal threat and left winger Dallas forced a fine save from Marcus Bettinelli after 19 minutes with a low drive from outside the box.
It was a warning to the Fulham defence and one they did not heed.
Just two minutes later Jake Bidwell once again combined with Dallas on the left and this time Dallas arrowed a drive from 20 yards that nestled into the bottom corner.
Seko Fofana had an effort blocked by Diagouraga while the lively Kacaniklic saw an instinctive first time left footed effort from 20 yards whistle inches over the bar.
After the break it was all Brentford and it was little surprise when a second goal arrived after 57 minutes.
Dallas’ first goal was good but his second was spectacular as Andre Gray teed him up and he lashed it into the top corner from 25 yards.
Fulham however were given hope in the 66th minute as a penalty was awarded very much against the run of play.
James Husband was upended in the box by Diagouraga and #11m striker Ross McCormack fired home from the spot for his 13th of the season.
Fulham had a decent shout for another penalty on 85 minutes when substitute Matt Smith went down after a clumsy challenge from James Tarkowski but Judge sealed the victory with a beautiful free-kick into the top corner from 25 yards in the 90th minute.
Deep in stoppage time, Jota added a fourth from Judge’s cross.