In an end-to-end game, Maxi Rodriguez scored an early brace to give the visitors a 2-0 lead, before stand-in goalkeeper Doni was dismissed for a foul on Junior Hoilett.
Rovers came back into the game with a Yakubu effort before the break, and the Nigerian attacker equalised from the penalty spot on the hour mark.
However, with the game heading for a draw, under-fire striker Andy Carroll hit the back of the net in injury time to give Liverpool the win and end their poor run of form of late.
Kenny Dalglish was happy with his team’s efforts after the game, but refused to discuss the referee’s performance after sending the Reds’ goalkeeper off.
“We’re not going to discuss the decision of the referee,” he confirmed to Sky Sports.
“We are not going to discuss it now. I don’t intend to discuss it.
“To get a man sent off and come away from here with a victory speaks great credit for them. It’s been a smashing day for us.
“We’re just delighted with the victory and we’re just going to give the boys all the credit they deserve.
“They’ve had a bit of abuse recently so let’s give them a bit of credit,” he concluded.
By Gareth McKnight