The second-from-bottom Reds cancelled out Joe Rothwell's 24th-minute opener moments before half-time when Carlton Morris stunned the high-flying hosts by netting an equaliser.
But it was largely one-way traffic after the restart with Ben Brereton-Diaz's 20th goal of a predatory season earning Rovers a sixth successive league win with Barnsley without an away victory since April.
The scoreline was flattering on Barnsley, with only some weak finishing from Rovers preventing a far more comfortable victory.
For the umpteenth time this season, loose play in midfield and poor game management from a side who are palpably lacking in Championship experience in the middle of the park, hurt Barnsley and it was something that Asbaghi has quickly found out from his time at the club already.
He said: "Obviously, I am disappointed. We created some great chances early on and then played into Blackburn's hands. We lost the ball to allow them to counter-attack.
"It was a good equaliser and we got more control in the second half, but that decisive moment came when we lost the ball and Brereton-Diaz attacked and scored lost us the game.
"Of course, we need to improve our offence and one way is to create more in counter-attacks and we scored a goal an improved in creating chances in this game, but it comes at a cost right now. We are creating more chances, but we have to make sure we have a good solid defence.
"Sometimes, we are doing it well, but at other times, we are losing balls in really dangerous areas. It is not until we mix the quality of being calm and creating chances in build-up from counter-attacks with being smart as to not lose the ball that we can actually win games.
"Yes, we are improving in creating chances and being more comfortable on the ball. But as a whole, we have not reached the ball where we are doing well in all the parts to win the game."