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The key moment for the visitors arrived in the 70th minute when Gareth Barry raced past Lucas Neill and pulled the ball back invitingly for Dion Dublin who screwed his shot disappointingly wide when unmarked.
At the other end, Andy Cole should also have hit the target when substitute Tugay had slipped him through but the ball bobbled and he blasted his effort into the crowd.
After Peter Enckelman had made a stunning save to tip a Cole header on to the bar in the first minute, the opening half became devoid of worthwhile chances as neither team looked good enough to score.
This lasted until a moment of magic from David Thompson set up Garry Flitcroft to thump an effort well over the top.
Juan Pablo Angel missed a handful of free headers to let Rovers off the hook and Barry went close with a whipped free kick as the half drew to a close.
In the second half, Cole worked space on the left but his low cross just by-passed the three Blackburn attackers and Damien Duff's deflected effort was kicked off the line by a wrong-footed Enckelman.
Duff caused more problems when he darted in from the left and his skidding effort beat Enckelman but also the far post.
Dublin headed over from substitute Stefan Moore's cross before the former Manchester United striker was replaced by Peter Crouch.
Souness said afterwards: "I can only blame our midweek game, we looked tired. Too many of our players were not at their best today."
His opposite number Graham Taylor reflected: "Our low scoring record is becoming a bit of a psychological problem. We had chances but we can't seem to put the ball in the back of the net."