Fulham scored four away from home for the first time since their 4-1 win over Crystal Palace in October 2013.
Moussa Dembele opened the scoring after just 100 seconds after a great ball from the left flank by James Husband.
The Whites were content to sit back and let the Robins attack before looking to pick them off on the counter. This tactic worked on 18 minutes but not in the way Kit Symons would have anticipated.
Aden Flint gave the ball away to Ryan Tunnicliffe. The midfielder laid in Cairney, who in turn set up Dembele for his second of the afternoon.
The third, coming after 33 minutes, was the goal of the game as Ross McCormack powered home a free kick from 30 yards after Luke Freeman felled James Husband.
Tunnicliffe added a fourth three minutes later when he lobbed Frank Fielding in the Robins' goal.
The Whites were content to settle for the three points and cruised through the second half, although Andy Lonergan had to be alet to tip Derrick Williams' shot over the bar in the closing stages.
There was concern though for Symons as Husband was forced off with what appeared to be an ankle injury and it was spoilt further as Jonathan Kodija scored a late consolation.
