The Reds raced into a 2-0 lead inside 16 minutes thanks to Maxi Rodriguez scoring twice but then things started to go wrong.
Alexander Doni, deputising for the suspended Jose Reina, was red-carded for bringing down Ayegbeni Yakubu and although third-choice goalkeeper Brad Jones saved the spot-kick with his first touch on his Reds Premier League debut, the Nigerian was not to be denied as his two goals either side of half-time levelled things up.
It appeared Rovers were set for a point which would take them out of the relegation zone only for Andy Carroll to head in the winner in the second minute of added time.
"We can't speak highly enough of the effort and commitment of the players. We went 2-0 up and then went down to 10 men," Dalglish said.
"The supporters drove them forward and in the second half they came out and did really well.
"At the end of the day we just about deserved to win. We are delighted for the players - it has not been the best of times for them.
"It doesn't mean to say we have turned any corners but it gives everyone a lift."