Liverpool keeper Doni, in for the suspended Jose Reina, was dismissed for a foul on Junior Hoilett and was replaced by Brad Jones who saved Ayegbeni Yakubu's penalty.
Maxi Rodriguez had already scored twice to seemingly put Liverpool in control, but Yakubu pulled a goal back before half-time and then netted a second-half penalty, conceded by Jones, before Andy Carroll netted a dramatic winner.
Dalglish refused to answer any questions about the performance of referee Anthony Taylor.
"We're not going to discuss the decision of the referee," Dalglish told Sky Sports 1.
When pressed he added: "We are not going to discuss it now. I don't intend to discuss it."
"I'm not trying to be evasive, I'm trying to be practical and factually correct. Once I've seen them I will be better educated.
"There is no point asking about anyone's sending off or anything else until we have seen the coverage."
"We are delighted with the result and the three points and the only way we can handle the post-match press conference is to praise the players," he added.
Dalglish preferred to concentrate on the performance of his players.
He added: "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 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."
When asked about Jones' foul on Yakubu which led to the second penalty and to a yellow card for the Australian, Dalglish responded: "I don't think he deserved to be sent off. The referee made the right decision."
He also praised Jones, who as third-choice goalkeeper is now in line to play in Saturday's FA Cup semi-final, and Carroll for their contributions.
"I think Brad had a bit of a psychological advantage because the two (he and Yakubu) had been at Middlesbrough together," he said.
"It was a great first contribution in the Premier League to save a penalty, I am sure he will remember that.
"I think Andy is very confident and got what he deserved. He worked really hard and scored a fantastic header - I don't think confidence is a problem."
Jones drew on his time when he was a team-mate of Yakubu at Middlesbrough to help him guess which way the Nigerian would choose to shoot from the penalty spot.
"On Fridays we used to practise penalties a lot so I knew his style and what he does," he told Sky Sports 1.
Jones, whose girlfriend gave birth to a son Nico Luca a week ago, following the death of his son Luca from leukaemia in November, thanked the Liverpool supporters.
He said: "The supporters have been brilliant. It's been a difficult 18 months with my son passing away. That was for him."
Jones is in line to play for Liverpool in the FA Cup semi-final with Everton on Saturday and said of that game: "It 's a big game and now it's just about preparing for that."
Carroll was able to see the lighter side of the night's proceedings as he said: "It was a funny old game, wasn't it?
"In the end we got the win we came here for. We kept on attacking all the way through.
"Coming here and winning, it's a great lift for the lads to go to Wembley."
Carroll's goal, only a third league victory of 2012 and the return of full-back Glen Johnson for the first time since February's Carling Cup final were all positives from the evening.
"The victory will do everyone the world of good. We've not had the best of results recently and that gives everyone a lift," said Dalglish.
"It is always that much more enjoyable when you win a game with 10 men.
"They deserve a bit of enjoyment, everyone at the club.
"And we have two other players - Glen and Daniel Agger - who came through the game and that will have done them the world of good and we need to assess them and see if they are to be considered for the squad for Saturday."
Blackburn manager Steve Kean was unhappy Taylor and his assistant had missed what he claimed was a foul on Grant Hanley by Martin Skrtel in the run-up to Liverpool's winner.
Kean claimed Hanley would have got back to where Daniel Agger headed the ball on for Carroll to head in the winner.
He said: "It's cruel because we felt we had done enough (to get a point).
"I feel very aggrieved about the last goal.
"Grant had tried to get back up but he can't get back up because Skrtel's pulling him to the ground."
"I have looked at the referee's decision, I can understand why he has not sent the sub goalie off because when the ball has spun up I think the referee thinks it is going to go to Liverpool player, so it doesn't prevent a goalscoring opportunity," he said.
On the performance he added: "You can't afford to miss penalties but when we got the second one and we scored for the equaliser I thought it was all us and we were chasing the third goal.
"The only way Liverpool were going to get something was on the break or a set-play and we needed that whistle or flag to go up for a blatant foul."