Sukur's first two goals for Rovers inspired Graeme Souness's side to victory and they are now three points clear in seventh place - which would land them a Uefa Cup place at the end of the season.
Fulham actually started the game better, but four goals in just 18 minutes - either side of half time - saw Rovers gain an easy win.
French star Steed Malbranque, as so often has been the case this season, was at the heart of everything good Fulham did and he had their best effort on goal after 30 minutes when he tested Brad Friedel from 25 yards.
Malbranque again had a good chance two minutes later, but he could not keep his effort down.
Rovers then made their West London opponents pay for their lack of invention in the last third, when they took the lead after 36 minutes.
Andy Cole struck the bar with a great effort which was blocked by Martin Djetou, the ball then came out to Gary Flitcroft who crossed for David Dunn and his flick-on found the arm of Lee Clark and referee Graham Poll pointed to the spot.
Dunn stepped up and easily beat Maik Taylor to give Rovers the lead, although they did not have long to wait for the second as it came just six minutes later.
Lucas Neill swung the ball into the area for Sukur (pictured), who controlled the ball exquisitely on his thigh before turning past Steve Finnan and slipping his shot under Taylor.
Fulham were deflated and never looked like getting back into the game and it was Rovers who extended their lead further.
Cole was once again involved as he took the ball down on the edge of the area before sliding a great pass to Damien Duff who from an acute angle on the left side of the box, blasted the ball beneath Taylor - who should have done better.
Again Rovers pushed forward and made it 4-0 with some more wonderful finishing just two minutes later.
Sukur this time brought the ball down and spotted Cole on the edge of the area, he played it into the former Manchester United star and his flick on fell straight into the path of the Turkish legend, who stayed calm and instead of shooting first time took it in his stride and steadied himself before easily placing the ball past Taylor.
Again Fulham offered little in return, with only Malbranque offering any form of resistance.
He came close on the hour mark and then with just four minutes left blasted well wide from just eight yards - which summed up a miserable evening for Jean Tigana's lacklustre outfit.
Click below for match summary:
Fulham 0-4 Blackburn Rovers
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