James McFadden equalised for the visitors with a fine free-kick, but Dunn headed home El-Hadji Diouf's corner from close range in the 67th minute to seal a 2-1 victory which virtually guarantees Blackburn's Premier League survival.
Rovers are now 14 points clear of the relegation zone with seven games to play thanks to a fantastic display from their talismanic top-scorer.
But the night could have been even sweeter for Dunn had referee Mark Clattenburg not disallowed his effort just before half-time - with Michel Salgado adjudged to have committed a foul in the build-up - a decision Allardyce felt was harsh.
"The difference today, no doubt about that, was the outstanding quality of David Dunn's three goals - that should have been," the Blackburn boss said.
"It was only two but should have been three. That was the difference to us getting three points, his outstanding quality."
Of Dunn's first goal, Allardyce said: "It took the goalkeeper by surprise, because I don't think he (Hart) thought he was going to shoot.
"Before the ball had hit the floor off the first touch, he's volleyed it, it's bounced, it's in the back of the net and the keeper can't get his feet across.
"He is a very, very good goalkeeper as we know, but that ball was in the back of the net and it was because he took his shot so early. Then he sneaked in for the winner for us in the end."
The strike Dunn had disallowed was almost as emphatic as his first, and Allardyce believed Clattenburg was wrong to chalk it off by penalising Salgado.
"It was supposed to be for a foul on (Lee) Bowyer but if you look at it again, there has been much worse that has gone on today, and I think it was a great shame when you've had a finish as good as that," Allardyce said.
"Dunny could be sat with the ball signed, with a hat-trick ball, but it doesn't really matter now because we have won the game and he has scored two. He is very happy, as all the players are."
There appeared to be evidence to the contrary as the Blackburn team walked off at half-time, with Dunn and Salgado seemingly arguing with each other just moments after the disallowed goal.
Asked what the dispute was about, Allardyce said: "I don't know, I'm not sure what the interpretation was between English and Spanish.
"They were having a bit of a tussle in the dressing room, but we just calmed it down and said 'look, we'll sort it out after the game'.
"I don't think it was anything to do with the goal, I think it was more to do with something else, perhaps defensively what we should or shouldn't have done. But it's nice to have a little bit of friction now and again."
Allardyce took Dunn off in the last 10 minutes after a challenge from Barry Ferguson and stressed he would not be taking any gambles on the midfielder's fitness ahead of Sunday's east Lancashire derby at Burnley.
"He (Dunn) got severely treated right towards the end there with some quite naughty tackles, one from Stephen Carr, the other from Barry Ferguson, so we whipped him straight off as quick as we could.
"We'll see what he is like tomorrow and take it from there. But we won't risk him, even though he scored two today.
"If we feel there might be any doubt after his treatment tonight, we'll put him on the bench. But we'll wait and see."
Birmingham boss Alex McLeish acknowledged Dunn's role in his team's defeat, but felt Clattenburg had failed to spot two infringements that led to Blackburn's second goal.
"He scored the two goals and there is no doubt that Dunn is a good player," McLeish said.
"But they were gifts in my opinion and I also thought they had a helping hand in the decision which led to the corner.
"It was a clear foul on Roger Johnson (before the ball went out) - it would have been a foul anywhere in Europe or international football, because (Christopher) Samba is climbing all over the top of Roger.
"And I've just been told that the ball was just outside the (quadrant when the corner was taken), so it's a shame these things weren't spotted. They should be spotted by professionals of that calibre.
"Blackburn weren't in the game after they scored. We totally dominated the game and they scored against the run of play, the second goal."
Source: Team Talk
Source: Team Talk