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The Welsh outfit now sit behind second-placed Watford on goal difference, but with a game in hand.
Norwich remains 10 points clear at the top of the table as its bid for an immediate return to the Premier League.
Steven Cooper's Swans fell behind to Bradley Dack's 37th-minute goal, but levelled four minutes before the break as Andre Ayew netted a penalty to score for the 15th time this season.
That would prove to be the only shot on target that Swansea -- bidding to return to the top flight for the first time since its relegation in 2017-2018 -- could manage as Blackburn were unlucky not to find a winner.
Elsewhere, QPR beat Wycombe 1-0 to leave the Chairboys rooted to the foot of the table, 12 points from safety.
On Thursday, Wayne Rooney's Derby visits Barnsley, with the opportunity to move nine points clear of the bottom three.