City had looked to have broken the Championship side's resistance after Alvaro Negredo put them ahead just before half-time.
But Dann replied early in the second half after a mistake by Costel Pantilimon and City were left hanging on after Dedryck Boyata was sent off.
Rovers' performance was an uplifting one after all their troubles of recent years and the atmosphere as they chased a late winner reminded the club of what they have missed since relegation.
City, with a Capital One Cup semi-final against West Ham coming up next Wednesday, made seven changes and perhaps paid the price with a loss of intensity.
Yet they still fielded a side full of international quality.
Negredo, Edin Dzeko and David Silva still formed a formidable attacking trio while Fernandinho provided drive from midfield.
City's most notable change again came in goal as Pantilimon replaced rested England number one Joe Hart.
The reasons were understandable but questions could be asked of Pellegrini's continuing rotatation of the two keepers after an indifferent display from the Romanian.
Rovers also made a change in goal, as one of Hart's England predecessors in Paul Robinson returned for his first game for 13 months.
The 34-year-old has now fully recovered after suffering a blood clot on the lung last year and his initial reintroduction was gentle as the visitors made a sluggish start.
Rovers, despite leaving star striker Jordan Rhodes on the bench, were positive and Rudy Gestede, fresh from making his loan from Cardiff permanent, troubled Joleon Lescott early on, although his penalty appeal was rejected.
The frustration for Rovers was that they could not work Pantilimon, with Dann's free header from a corner their best effort but easily claimed by the keeper.
City slowly began to take control of the game midway through the first half and Robinson twice came out bravely to thwart Negredo.
Fernandinho shot straight at Robinson from outside the box and Silva spurned a glorious chance when he scooped over from a Boyata cut-back.
Tommy Spurr then appeared to pull back Negredo as James Milner lofted in a cross but there were few penalty appeals.
Rovers pushed again before the break but Gestede failed to connect with a headed chance and Chris Taylor shot over.
City punished them by snatching the lead themselves on the stroke of half-time as Dzeko flicked on a Silva corner and Negredo turned in his 15th of the season from close range.
The manner in which the goal was conceded perhaps led to what seemed a frank exchange of views between Grant Hanley and Taylor as the players left the field at the interval.
Rovers came back fired up and earned reward as they grabbed an eqauliser in the 54th minute.
Gestede rose high to meet a Ben Marshall corner with a powerful header and Pantilimon could not hold it. Dann rushed in to sweep past him from close range.
City looked troubled at the back and a long throw-in from Spurr gave them more difficulties.
The pace of the game noticeably increased as Taylor put another Blackburn chance over.
City responded with Dzeko testing Robinson from outside the area and Negredo heading wide from the rebound.
Navas came on for Negredo and brought another fine save out of Robinson, who also spilled but regathered another effort from Dzeko.
Blackburn were undeterred and sensed an opportunity to pull off a shock.
They sent on the additional firepower of Rhodes and DJ Campbell with nine minutes remaining and threw everything they could at City.
City's cause was not helped as Boyata upended Campbell to earn his second yellow card.
Rovers poured forward again but the impressive Tom Cairney sent a good shot narrowly wide from distance.
Lescott also needed to scramble back to break up another attack as City clung on.