Yakubu's heroics lifted Rovers off the bottom of the Premier League, but Kean was still jeered by some home fans, who continued to chant for his dismissal and protested against the manager after the final whistle.
Brendan Rodgers' Swans had hauled themselves back into the game with goals from Leroy Lita and Luke Moore, and were pressing for an equaliser at 3-2 down when Joe Allen was sent off.
Rovers are now in 18th place, one point and one place from safety after only their second league win of the season.
Yakubu grabbed his first in the 20th minute after superb work from Gael Givet, who cut in from the left flank and crossed for the Nigerian to blast home the opener.
But even that goal did not placate some sections of the home support, who booed when Kean celebrated with the goalscorer on the touchline.
It was clear Yakubu was up for the challenge, but after he sent another chance squirming wide, it was the visitors who notched an equaliser in the 35th minute.
Mark Gower sent in a looping cross from the right flank and Lita, in for the injured Danny Graham, rose to angle a fine header in off the post.
Yakubu restored Blackburn's lead in first-half injury time when he reacted with a point-blank header after substitute Simon Vukcevic volleyed across the box.
Scott Sinclair was at the centre of almost all Swansea's moves early in the second half, setting up Lita to blaze over the bar then unleashing a shot of his own into the side-netting.
Rovers goalkeeper Paul Robinson played a big part in Rovers' third when his booming goal-kick was tipped over the bar by opposite number Michel Vorm.
From the resulting corner Rovers grabbed their third, with Christopher Samba heading back Morten Gamst Pedersen's set-piece and Yakubu pouncing to finally complete his hat-trick from close range.
But Swansea were far from done and they reduced the deficit in the 66th minute when Moore fired home from inside the box after Robinson had parried an initial effort from Wayne Routledge.
With nerves clearly jangling among the Rovers rearguard, Sinclair was given space to blaze over the bar before Neil Taylor's foul on Vukcevic allowed Yakubu to complete his solo show.
With Allen having been shown a second yellow card seconds before the goal, Swansea's hopes were over.
Source: AFP
Source: AFP