Gylfi Sigurdsson produced a brilliant curler to grab his first home goal for Swansea and give them a 37th-minute lead, with Nathan Dyer doubling the advantage six minutes later. Scott Dann then capped a terrible day for Rovers by putting through his own net just after the hour mark.
Midfielder David Dunn hit the post with the visitors' best effort but this was a miserable display that leaves Steve Kean's side three points from safety in 19th place having played at least a game more than the sides around them.
Rovers' final four games include trips to Champions League hopefuls Tottenham and Chelsea, and their home games against Norwich and Wigan are likely to be vital if they are to retain their top flight status.
This was a fifth defeat in a row and the travelling support made their feelings towards manager Kean and club owners Venky's perfectly clear during the latter stages of this abject performance.
For Swansea victory ended a run of four defeats and all but guarantees them another season of Premier League football.
Rovers had started the better of the two sides and had a penalty appeal turned down inside the opening three minutes as Ayegbeni Yakubu claimed that Angel Rangel had clipped his heels when Junior Hoilett's miscued volley flew into his path.
Dann then headed over from a Morten Gamst Pedersen corner as the visitors had several chances to deliver into the Swansea box from set-pieces.
But once Swansea began to find some fluency on the ball they took control of territory and possession.
The hosts were convinced they should have had a penalty in the 23rd minute as Pedersen came in from behind on Sigurdsson as the midfielder prepared to connect with Rangel's ball across the box, but referee Mark Clattenburg waved away the appeals.
The majority of the game was being played in the Blackburn half, and with Yakubu failing to give them an outlet chances began to flow Swansea's way.
Rovers' captain Paul Robinson just about managed to keep out a ferocious, swerving drive from the dangerous Sigurdsson but the former England keeper was beaten by the Iceland international seconds later.
Joe Allen floated a lovely ball across the box which Sigurdsson brought down to curl a left-footed effort into the far corner of the net for his seventh goal since his loan move from Hoffenheim.
Blackburn were rocked and an awful pass from Steven Nzonzi played Danny Graham in, but Robinson made a fantastic sprawling save to his left to keep the striker's low shot out.
But the lead soon doubled as Dann's woeful clearance fell to Scott Sinclair. The winger played the ball inside to Graham, who in turn unselfishly set up Dyer for the winger to slice the ball past Robinson.
Dunn had a decent chance to pull a goal back for Rovers shortly after the break but he scuffed his volley from Dann's header back across goal.
And normal service was soon resumed as patient Swansea passing ended with Rangel firing over from just inside the area.
Having sent on Mauro Formica for Grant Hanley at the interval, Kean then removed Hoilett in favour of Anthony Modeste, and David Goodwillie followed in place of Yakubu.
But the changes had little impact as Swansea extended their lead further in the 63rd minute.
Sinclair danced his way into the box, beating three defenders before teeing up Sigurdsson. The midfielder's effort hit the post but the unfortunate Dann directed the rebound into his own net.
Dunn then headed over from inside the six yard box, and then struck the post in a brief Rovers rally, before Robinson prevented Sinclair making it four with a save low to his right.
From that point Swansea were able to cruise to victory in comfort.
They could have added further goals but Sinclair fired over Robinson's bar after being allowed to easily cut in from the left, while Robinson then made another agile save in the dying moments to prevent substitute Leroy Lita finding the net.