Both these sides had suffered 3-1 defeats in the week, and only administration-hit Sheffield Wednesday were below the hosts in the table at kick-off.
Southampton forward Adam Armstrong – yet to score against former employers Rovers in five attempts – combined well with Finn Azaz early on to set up Cameron Archer and it took an important block from Balazs Toth in the home goal to keep the game goalless.
At the other end, Taylor Gardner-Hickman hit the woodwork direct from a corner, and Yuki Ohashi was denied inside the area by a block from Joshua Quarshie.
The opener came from a mistake in the Rovers back-line – a misplaced pass allowed Scienza to intercept possession, and the Brazilian stayed cool to fire home a left-footed opener for his first goal since moving to the south coast from German side Magdeburg in the summer.
Blackburn came out stronger after the break, and Ohashi found another Saints player between him and the net in the form of visiting stopper Alex McCarthy, who was alert to deny the striker from inside the six-yard box.
Ryoya Morishita also wasted a good chance to equalise when he fired wide when one-on-one with McCarthy, while at the other end, Toth denied Armstrong twice, the second time using his feet to divert the ball onto the post.
A frantic final 15 minutes saw Rovers snatch victory from the jaws of another home defeat, starting with right-back Alebiosu, whose first goal in senior football came from a direct cross from the right that evaded everyone and nestled in the corner.
Ten minutes later, Rovers had all three points in their hands as Gudjohnsen fired past McCarthy after the stopper had denied Ryan Hedges from close range.
In finding the net, Gudjohnsen achieved what his father Eidur never could during his professional career, a goal at Ewood Park.
A bad final few minutes for Southampton was made worse when substitute Welington received a red card for elbowing match-winner Gudjohnsen in the centre circle.
