Scottish international midfielder Quashie made a highly impressive debut following his arrival in a £1.4million move from Southampton as first-half goals from skipper Kevin Campbell and Jonathan Greening earned Albion a win that has given their survival hopes a major boost.
With Quashie running the show in midfield, Albion secured only their second win in eight games to keep themselves clear of the relegation zone.
It was a victory against the odds as Blackburn were the Premiership's form team before they arrived in the Midlands.
They had won five and drawn one of their previous six games - taking 16 points from a possible 18.
But they were a pale shadow of the side who had fired four goals past Manchester United in their previous game as their hopes of qualifying for Europe next season suffered a setback.
They never recovered from Campbell's seventh minute opener and their fate was sealed when Greening struck 13 minutes before half-time.
Campbell should have fired Albion ahead seconds before his opener as he missed a golden chance in the sixth minute.
The former Arsenal and Everton marksman was unmarked at the near post, but somehow conspired to fire wide after a cross from Greening.
Campbell must still have been cursing his poor finishing when the chance to make amends arrived 60 seconds later.
Blackburn skipper Andy Todd was powerless to cut out a raking crossfield ball from Junichi Inamoto. Campbell collected it in his stride and took a couple of paces forward before drilling his shot past goalkeeper Brad Friedel.
It was a setback that Blackburn struggled to get to grips with and they would have fallen further behind three minutes later only for Friedel to pull off an acrobatic save to deny Greening who let fly from 25 yards.
When Blackburn did pose a threat of their own, they lacked a cutting edge with David Bentley struggling to reproduce the form that had seen him score a hat-trick against United.
In a flurry by Blackburn, Steven Reid had a shot blocked by Paul Robinson and Morten Gamst Pedersen saw a free-kick turned away by goalkeeper Tomasz Kuszczak.
They were made to pay again for their lethargic display when Greening doubled Albion's lead.
He cashed in on a misplaced clearance from left-back Michael Gray that flew back towards his own penalty area.
Greening collected the ball and held off the challenge of Tugay before firing a 25-yard effort past Friedel.
That secured the win for Albion with Blackburn wasting one golden chance to force their way back into the game.
Reid's 68th minute centre caught the Albion defence flat-footed only to be missed by Robbie Savage on the six-yard line.
When the ball found its way to the unmarked Pedersen at the far post he could only steer his header across the face of goal. It really was one of those days for Blackburn.