Before the goals, no fewer than three shots had hit the woodwork in an exciting opening spell.
Rovers duo Andy Cole and Henning Berg were both unlucky not to bag the opener, while Fulham forward Steve Marlet also hit the post. Hayles pounced in the 32nd minute, when referee Mark Halsey made an excellent decision to play advantage rather than halt a sweeping move for a penalty after Henning Berg's handball. Hayles hammered home the loose ball moments after Berg's offence.
The second period was also packed with attacking invention, mainly from the home side.
From one such move, Malbranque played a fine long-range one-two with Marlet and thumped home a great volley to clinch a 2-0 victory.
Rovers' woe was complete when Craig Short was sent off in injury time after elbowing Marlet in the face.
England hopeful David Dunn returned to the Blackburn first XI after a three-week lay-off with a calf injury.
He plugged the gap left by suspended skipper Garry Flitcroft. Cottagers midfielder Sean Davis was back in Jean Tigana's team after his broken nose.
Referee Halsey was a busy man in the opening stages as Fulham's Malbranque earned himself a needless booking inside the opening 60 seconds when he blocked Lucas Neill's quick free-kick just a yard from the ball.
And Malbranque's team-mate Andy Melville also went into the official's notebook a minute later when he impeded Cole on the visitors' right wing.
Halsey also brandished his yellow card at Rovers' Lucas Neill after his late tackle on Brevett.
Cole came agonisingly close to the opener in the sixth minute when he raced into clear space between Cottagers centre-backs Melville and Alain Goma.
The former Manchester United striker's 14-yard effort thumped against Edwin van der Sar's right-hand post and rebounded into the grateful goalkeeper's hands.
Van der Sar pulled off a sensational instinctive block from a close-range Berg volley.
The centre-back met Damien Duff's corner with a sweet strike - but the Dutch keeper made a wonderful reaction save to divert the shot onto the face of his bar. The ball rebounded to safety.
Next, it was the home side's turn to hit the woodwork, when Malbranque's complete miskick fell to Steve Marlet inside the Blackburn penalty area.
The £11.5million striker, with his back to goal, flicked the ball up and swivelled to hook it against Brad Freidel's right-hand upright.
Fulham took the lead after 32 minutes, thanks to that excellent piece of refereeing by Halsey.
Marlet robbed Tugay 30 yards from the home side's goal, raced to the other end of the field and played a great one-two with Hayles.
Marlet's cross from the right hit Berg's arm and Halsey's linesman flagged for a penalty.
However, the referee allowed play to continue under the advantage rule and Hayles swooped to ram the loose ball into the Rovers net from seven yards.
Freidel made an excellent diving save from a 20-yard Malbranque free-kick as the interval neared - and shortly after the break, Dunn wasted a great chance to draw level as he raced onto Duff's curling cross but headed harmlessly wide.
Fulham midfielder Malbranque established a two-goal cushion for the home side with a fine goal from a lightning breakaway in the 63rd minute.
The France Under-21 international collected Legwinski's clearance just inside his own half and threaded a ball out to Marlet on the right wing.
He delivered a fine deep cross which Malbranque raced 50 yards to meet with a sweet volley that bounced just in front of Freidel and into the net via his left hand.
Freidel made amends for his role in the second goal with a marvellous save after another swift home break ended with Marlet's thumping close-range volley.
Van der Sar prevented Rovers pulling one back with a solid save from Craig Hignett's 25-yard drive - and Cole fluffed an absolute sitter from eight yardsto sum up Blackburn's inept attacking.
Rovers' woe was complete when Short was sent off in injury time after elbowing Marlet in the face.